Inference

Question 1

Sample_size = 1250

p=450/1250

What is the 95% confidence interval estimate for the population proportion?

Answer 1

• p = 450/1250 = 0.36*
• The critical z-scores for 95% are ±1.96*
• So the answer is: 0.36 ± (1.96*sqrt(0.36*0.64/1250))*

Question 2

A country is voting to choose a President; there are two candidates A and B. Every citizen has already decided if they are going to vote for A or B. You conduct a sample survey to predict the election result. What distribution will you use to model this problem?

Answer 2

A Bernoulli distribution

Question 3

A country with a 100 million voters is voting for an election with 2 candidates A and B. A random sample of 100 voters shows that 57 people vote for And 43 people have voted for B. What is our estimate of the population variance in terms of B?

Answer 3

s² = (57*(0-0.43)^2 + 43*(1-0.43)^2)/(100-1)

= 0.24757575757

Question 4

We want to compare the effect of access to computers on student performance in a large high school. Two random samples of 10 students each are drawn; one sample with computers and another same without. A t-test is used to compare the GPAs of the two groups. Which of the following is a necessary assumption?

(A) The population standard deviations from each group is known.

(B) The population standard deviations from each group are unknown.

(C) The population standard deviation from each group are equal.

(D) The population of GPA scores from each group is normally distributed.

(E) The samples must be independent samples and for each sample np and n(1-p) must both be at least 10.

Answer 4

(D) Since the sample sizes are small, the sample must come from normally distributed populations. The sample should be independent; np and n(1-p) refer to conditions for test involving sample proportions, not means.

Question 5

How does doubling the sample size change the confidence interval size?

(A) the interval size.

(B) Halves the interval size

(C) Multiplies the interval size by 1.414

(D) Divides the interval size by 1.414

(E) This question cannot be answered without knowing the sample size.

Answer 5

(D) If the sample sizes in increased by a factor d, the interval estimate is divided by sqrt(d)

Question 6

Population mean estimate is 9250 with standard deviation =2575. A random sample of size 50 is found to have a sample mean more than 500 different from the population mean estimate. What is the probability we will mistakenly reject a true claim?

(A) 0.043

(B) 0.085

(C) 0.170

(D) .830

(E) .915

Answer 6

(C)

H₀: µ=9250

H_a: µ≠9250

Standard deviation of the sample means is: 2575/sqrt(50) = 364.2. (Should we not be using the sample standard deviation?)

The critical z-scores are ± 500/363.2 = ±1.373

alpha - P(x ≤ -1.373) + P(x ≥1.373) = 1 - normalcdf(-1.373,1.373) =.170.

There is a 17% chance of a Type I error.

Question 7

A 2007 survey of 980 American drivers concluded that 38% percent of the driving population would be willing to pay higher gas prices to protect the environment. Which of the following best describes what is meant by the poll having a margin of error of 3%?

(A) Thre percent of those surveyed refused to participate in the poll.

(B) It would not be unexpected for 3 percent of the population to readily agree to the higher gas price.

(C) Between 343 and 402 of the 980 drivers surveyed responded that they would be willing to pay higher gas prices to protect the environment.

(D) If a similar survey of 980 American drivers was taken weekly, a 3% change in each week’s results would not be unexpected.

(E) If is likely that between 35% and 41% percent of the driving population would be willing to pay higher gas prices to protect the environment.

Answer 7

(E) Using a measurement from a sample, we can never say exactly what the corresponding population proportion is. We can say that we have a certain confidence that the population proportion lies in a particular interval. (I think we can say that we are 95% certain that the answer lies between 38±3 percent.

Question 8

A doctor wants to know if the number of patients he sees is related to the day of the week. Write out the expression for the expected value of the chi-square for the appropriate test?

Mon : 12, Tue: 5, Wed: 9, Thu: 4, Fri: 15

Answer 8

The expected value for each cell for a uniform distribution is:

(12+5+9+4+15)/5 = 45/5 = 9

The chi-square = ∑((obs - exp)²⁾/exp

= ((12-9)^2 + (5-9)^2 + (9-9)^2 + (4-9)^2 + (15-9)^2)/9

Question 9

What is the chi-squared test?

Answer 9

A chi-squared test, also referred to as 𝛘² test(or chi-square test), is any statistical hypothesis test in which the sampling distribution of the test statistic is a chi-square distribution when the null hypothesis is true.

Question 10

What is the difference between a numerical variable and a categorical variable?

Answer 10

ategorical variable yield data in the categories and numerical variables yield data in numerical form. Responses to such questions as “What is your major?” or Do you own a car?” are categorical because they yield data such as “biology” or “no.” In contrast, responses to such questions as “How tall are you?” or “What is your G.P.A.?” are numerical. Numerical data can be either discrete or continuous. T

Question 11

Define: standard independent normal variable

Answer 11

The simplest case of a normal distribution is known as the standard normal distribution. This is a special case when μ=0 and σ=1, and it is described by this probability density function:

Phi(x) = (e^(-1/2*x)^2)/sqrt(2*π)

Question 12

Why is the chi-squared test called the chi-squared?

Answer 12

1. The chi-square is a reference to the square of the Greek letter chi which looks like X².
2. The X actually refers to the x as an independent normally distributed random variable.
3. In other words: if you take take a independent normally distributed random variable and square it, you will get a X² which follows its own distribution with degree of freedom 1.
4. For example:

chi_square_3_degrees = (X₁)² + (X₂)² + (X₃)²

where X1, X2 and X3 and three different normally distributed random variables.

Question 13

A restaurant owner thinks that he gets customers in the following distribution:

Mon: 10%, Tue:10%, Wed:15%, Thu:20%, Fri:30%, Sat:15%.

The restaurant is closed on Sunday. Sal keeps track of the data for 1 week and gets the following numbers:

Mon: 30, Tue: 14, Wed: 34, Thu: 45, Fri: 57, Sat: 20.

Is the restaurant owner correct, setting alpha=5%?

Answer 13

Observed: Mon: 30, Tue: 14, Wed: 34, Thu: 45, Fri: 57, Sat: 20

Expected: 10% 10% 15%, 20% 30% 15%

Expected(abs): 20 20 30 40 60 30

X² = ((30-20)^2)/20 + ((14-20)^2)/20 + ((34-30)^2)/30 + ((45-40)^2)/40 + ((57-60)^2)/60 + ((20-30)^2)/30

= 136/20 + 16/30 + 25/40 + 9/60 + 100/30 ≈ 11.44

Degrees of freedom = 6-1=5

Critical value for 5 degrees of freedom and alpha =5% is: 11.07.

Since 11.44 is more extreme than 11.07, we reject the null hypothesis that the owner is correct.

Question 14

A survey of 1000 Americans reveals that 525 believe that whales are an endangered animal; in a survey of 750 Japanese, 325 believe that they are endangered. To test at 5% significance level whether or not the data are significant evidence that the proportion of Japanes who believe that whales need protection is less than the proportion of americans with this belief, a student sets up the following:

H₀: p=0.525 and H_a: p

Which of the following is a true statement?

(A) The student has set up a correct hypothesis test.

(B) Given the large sample sizes, a 1% percent significance level would be more appropriate.

(C) A two sided test would be more appropriate.

(D) Given that (525+325)/(1000+750) = .486, Ha: p

(E) A two-population difference in proportions hypothesis test would be more appropriate.

Answer 14

(E) We are comparing two population proportions and thus the correct test involves: Ho: p1 - p2 = 0 and Ha: p1 - p2 > 0 where p1 and p2 are the respective proportions of Americans and Japanes who believe that whales need protection.

Question 15

What is the probability of a Type II error when a hypothesis test is being conducted at the 5 percent significance level (alpha = .05)?

(A) .05 (B) .10 (C) .90 (D) .95

(E) There is insufficient information to answer the question.

Answer 15

(E) There is a different probability of Type II error for each possible correct value of the population parameter.

Question 16

A fitness center advertises that the average pulse rate of its members is 68.4 bpm.

A SRS of 48 members, find a mean of 71.0 bpm with a standard deviation of 10.3 bmp. In which of the following interval is the P-value located?

(A) P

(B) .01

(C) .02

(D) .05

(E) P > .1

Answer 16

(D) Ho: µ = 684, Ha = µ = 68.4.

σ_s = 10.3/sqrt(48) = 1.487.

t-score for 71.0 is (71 - 68.4)/1.487 = 1.748

With df = 48 - 1 = 47 we have P/2 = tcdf(1.748, 1000, 47) = .0435 and P=.087

Question 17

A guidance counselor wishes to determine the mean number of changes in academic major by college students to within ±0.1 at a 90% confidence level. What sample size should be chosen if it is known that the standard deviation is 0.45?

(A) 8 (B) 54 (C) 55 (D) 78 (E) 110

Answer 17

(C).

1.645*(0.45/sqrt(n)) ≤ 0.1

sqrt(n) ≥ 7.4 and n ≥ 54.8

So choose n = 55

Question 18

For a one-sided hypothesis test for the mean, for a random sample of size 15, the t-score of the sample mean is 2.615. Is this significant at the 5 percent level? At the 1 percent level?

(A) Significant at the 1 percent level but not at the 5 percent level.

(B) Significant at the 5 percent level but not at the 1 percent level.

(C) Significant at both the 1 percent and 5 percent levels.

(D) Significant at neither the 1 percent nor 5 percent levels.

(E) Cannot be determined from the given information.

Answer 18

(B)

With df = 15 -1=14, the critical t-scores for the .05 and .01 tail probabilities are 1.761 and 2.624 respectively.

We have 2.615 > 1.761 but 2.615

Alternatively: P(t > 2.615) = tcdf(2.615,1000,14) = .0102

.0102 .01

Question 19

A confidence interval estimate is calculated from the data of a random sample of size n. All other things being equal, which of the following will result in a smaller margin of error?

(A) A greater confidence interval

(B) A larger sample standard deviation

(C) A larger sample size

(D) Accepting less precision

(E) Introducing bias into sampling

Answer 19

C

The margin of error varies:

1. directly with the critical z-value and
2. directly with the sigma of the sample
3. inversely with square root of the sample size.

Question 20

Which of the following is true:

(A) Tests of significance (hypothesis tests) are designed to measure the strength of evidence against the null hypothesis.

(B) A well-planned test of significance should result in a statement that the null hypothesis is true or false.

(C) The null hypothesis is one-sided and expressed using either if there is interest in deviations in only one direct.

(D) When a true parameter value is farther from the hypothesized value, it becomes easier to reject the alternative hypothesis.

(E) Increasing the sample size makes it more difficult to conclude that an observed difference between observed and hypothesized values is significant.

Answer 20

A

1. We attempt to show that the null hypothesis is unacceptable by demonstrating that it is improbable.
2. We cannot show that it is definitely true or false.
3. If the interest in deviations in only one direction, the alternative hypothesis is expressed using either .
4. When a true parameter value is farther from the hypothesized value, it becomes easier to reject the null hypothesis.
5. Increasing the sample size makes it easier to conclude that the observed difference between observed and hypothesized values is significant.

Question 21

A geneticist claims that four species of fruit flies should appear in the ration 1:3:3:9. Suppose that a sample of 480 flies contained 25, 92, 68 and 295 flies of each species, respectively. Does a chi-square test show sufficient evidence to reject the geneticist’s claim?

(A) The test proves the geneticist’s claim.

(B) The test proves the geneticist’s claim is false.

(C) The test does not give sufficient evidence to reject the geneticists’s claim.

(D) The test gives sufficient evidence to reject the geneticists’s claim.

(E) The test is inconclusive

Answer 21

D

1. Expected values are 30, 90, 90, 270
2. X² = ((25-30)^2)/30 + ((92-90)^2)/90 + ((68-90)^2)/90 + ((295-270)^2)/270 = 8.57
3. With df = 4-1=3, the P-value is .0356
4. On the TI-84:
   
   1. Put {25, 92, 68, 295} in L1 and
   2. {30, 90, 90, 270) in L2
   3. X²GOF-Test

Question 22

Two confidence interval estimates from the same sample are {72.2, 77.8} and {71.3, 78.7}. One estimate is at the 95% level and the other is at the 99% level. Which is which?

(A) {72.2, 77.8} is the 95% level.

(B) {72.2, 77.8} is the 99% level.

(C) The question cannot be answered without knowing the sample size.

(D) The question cannot be answered without knowing the sample standard deviation.

(E) This question cannot be answered without knowing both the sample size and standard deviation.

Answer 22

A

The narrower interval corresponds to the lower confidence level.

Question 23

We want to compare the SAT math and verbal scores of college applicants. A SRS of 40 applicants is chosen and the math and verbal scores are noted. Which of the following is the proper test?

(A) Test of difference in two population means.

(B) Test of difference in two populaton proportions.

(C) One sample test on differences in paired data.

(D) Chi-square goodness-of-fit tes.t

(E) Chi-square test for homogeneity

Answer 23

C

1. Two sample tests require that the samples being compared be independent of each other.
2. In this case, the SAT verbal and math scores are of each individual applicant.
3. The proper procedure is to run a one sample test on the single variable consisting of the difference between math and verbal SAT scores for each application.

Question 24

To test whether husbands or wives have greater manual agility, an SRS of 50 married couples is chosen and all 100 people are given a scored test. What is the conclusion at a 5% significance level if a two-sample hypothesis test, H_o: µ₁ - µ₂=0, H_a: µ₁ - µ₂ = 0, results in a P-value of .15.

(A) The observed difference between husbands and wives is significant.

(B) The observed difference is not significant.

(C) A conclusion is not possible without knowing the mean scores placed by husbands and by wives.

(D) A conclusion is not possible without knowing both the mean and standard deviation of the scores by husbands and by wives.

(E) A two-sample hypothesis test should not be used in this example.

Answer 24

E

The two sample hypothesis test should be used only when the two sets are independent. In this case, there is a clear realationship between the data, in pairs.

The proper procedure is to run a one-sample test on the single variable consisting of the differences from the paired data.

Question 25

Null hypothesis in the morning: the weather will remain dry. What would be the results of Type I and Type II errors.

(A) Type I error: get drenched. Type II error: endlessly carry around an umbrella.

(B) Type 1 error: endlessly carry around an umbrella. Type II error: get drenched.

(C) Type 1 error: carry an umbrella and it rains. Type II error: carry no umbrella, but weather remains dry.

(D) Type I error: get drenched. Type II error: carry no umbrella, but weather remains dry.

(E) Type I error: get drenched. Type II error: carry an umbrella, and it rains

Answer 25

B

Type I error means the null hypothesis is correct (the weather will remain dry) but you reject it (thus you needlessly carry around an umbrella).

A Type II error means that the null hypothesis is wrong (it will rain) but you fail to reject it (thus you get drenched).

Question 26

A political scientist thinks that the percentage of people holding racist views is higher than the previously claimed 7%. A random sample of 300 people finds 23 people holding these racist views. Is this strong evidence agains the 7% claim?

(A) Yes, because the P-value is .0062

(B) Yes, because the P-value is 2.5.

(C) No, because the P-value is only .0062

(D) No, because the P-value is over .10

(E) There is insufficient information to reach a conclusion

Answer 26

A

1. H_o: p=.07 and H_a: p > .07.
2. sigma_phat = sqrt( (.07)*(.93)/200 ) = 0.01804161855
3. p_hat = 23/200 = .115 with a z-score = (.115 - .07)/.018 = 2.5
4. P-value = .0062
5. On the TI-84 calculator, use the 1-PropZTest

Question 27

A political scientist thinks that the percentage of people holding racist views is higher than the previously claimed 7%. A random sample of 300 people finds 23 people holding these racist views. We want to find out if there is strong evidence agains the 7% claim.

Which function will you use on the TI-84 calculator and what are its parameters?

Answer 27

1-PropZTest

Parameters:

1. p0: The hypothesized proportion
2. x: number of favorable outcomes,
3. n: the sample size
4. Then select the alternate hypothesis.

Question 28

In sampling 200 people, we found that 30% of them favored a certain candy. Use α = 10% to test the hypothesis that the proportion of people who favored that candy is less than 35%.

Demostrate how we can use the TI-84 to solve this problem.

Answer 28

See examples at: http://cfcc.edu/faculty/cmoore/TI83ZTest.htm

1. ### This represents a one-sample test of proportion.
2. ### So we use the “1-PropZTest” function.
3. ### The sample proportion is 30% or p = 0.30, and
4. ### the hypotheses are H₀: p ≥ 0.35 and H₁: p
5. ### Hypothesized value is 0.35.

Press STAT and the right arrow twice to select TESTS.

Use the down arrow to select
5:1-PropZTest…

Press ENTER.

1. Enter hypothesized proportion,
2. number of favorable outcomes, x,
3. sample size, n, and
4. select the alternate hypothesis.

Use down arrow to select Calculate and press ENTER.

Results:

Since the p = 0.069 is less than α = 0.10, reject the null ypothesis. Conclude that the sample proportion of 0.30 is significantly less than the hypothesized proportion of 0.35.

Question 29

A population has a bell-shaped histogram with a mean of 50 and a s.d. of 5; we represent the population of values by a normal distribution.

We are interested in the percentage of this population with values between 45 and 60. This percentage is represented by the area under the curve between 45 and 60. Since there are an infinite number of possible normal distributions, this may seem like an impossible task, but this family of distributions has a property that makes this feasible. What is that property?

Answer 29

Closure under linear transformations.

This property is formally called closure under linear transformations.

If a population has a normal distribution with some mean µ and s.d. σ , and if we multiply each value in this population by a constant a and then add a constant b, then these modified values also will have a normal distribution with mean a+bµ and with s.d. |b|σ .

Note that the s.d. of the new values is the absolute value of the slope times the original s.d.

Question 30

A population has a normal distribution with mean µ and s.d. σ, and if we multiply each value in this population by a constant a and then add a constant b, then these modified values also will have a normal distribution with mean ____ and with standard deviation ____.

Answer 30

A population has a normal distribution with mean µ and s.d. σ, and if we multiply each value in this population by a constant a and then add a constant b, then these modified values also will have a normal distribution with

mean a+bµ and

with s.d. |b|σ.

This property is formally called closure under linear transformations.

Note: we generally use the symbol µ for the sample mean and the symbol σ for the sample standard deviation. I am not sure why the author has used them for the population mean and population standard deviation.

Question 31

For problems involving Hypothesis Test of Mean for Normal Distribution (Sigma, σ, is Known) - One Sample. What is the procedure on the TI-84 calculator?

Answer 31

1. Press STAT and the right arrow twice to select TESTS.
2. Select the highlighted 1:Z-Test option.
3. Use right arrow to select Stats (summary values rather than raw data) and Press ENTER.
4. Use the down arrow to Enter the hypothesized mean, population standard deviation, sample mean,and sample size.
5. Select alternate hypothesis.
6. Press down arrow to select Calculate and press ENTER.

Question 32

A sample of size 200 has a mean of 20. Assume the population standard deviation is 6. Use the TI-83 calculator to test the hypothesis that the population mean is not different from 19.2 with a level of significance of α = 5%.

Answer 32

1. Use the Z-Test under Stats.
2. Enter the relevant data.
3. Since the p-value is 0.1, do not reject the null hypothesis with an α(alpha) value of 0.10 or smaller (10% level of significance or smaller). [In this example, α = 0.05.]

See: http://cfcc.edu/faculty/cmoore/TI83ZTest.htm

Question 33

Which test do we use on the TI-84 calculator for:

Hypothesis Test of Mean for Normal Distribution (Sigma, σ, is Known) - Two Sample

Answer 33

2-SampZTest

1. Use right arrow to select Stats (summary values rather than raw data).
2. Enter standard deviations, mean and sample size for samples 1 and 2.
3. Select alternate hypothesis.
4. Press down arrow to select Calculate and press ENTER.

Question 34

In sampling 200 freshman college students (Sample 1), we found that 61 of them earned an A in statistics. A sample of 250 sophomore college students (Sample 2) had 60 people who earned an A in statistics. Test the hypothesis that the proportion of freshmen that earned an A in statistics is greater than the proportion of sophomores that earned an A in statistics.

Which procedure do we use on the TI-84 calculator?

Answer 34

2-PropZTest

Solution: This represents a two-sample test of proportion. We use the “2-PropZTest” function. The hypotheses are H₀: p₁ ≤ p₂ and H₁: p1 > p2 (claim).

Results:

Since the p = 0.061, reject the null hypothesis for values of a > 0.061. Conclude that the sample 1 proportion of 0.305 is significantly greater than sample 2 proportion of 0.24 when a > 0.061.

Question 35

if a set of data has a histogram that is approximately bell-shaped, then approximately ___% of the measurements are within 1 standard deviation of the mean, approximately ___% are within 2 standard deviations of the mean, and approximately ___% are within 3 standard deviations of the mean

Answer 35

if a set of data has a histogram that is approximately bell-shaped, then approximately 68% of the measurements are within 1 standard deviation of the mean, approximately 95% are within 2 standard deviations of the mean, and essentially all (makes more sense than approximately 99.74%) are within 3 standard deviations of the mean.

From UT Dallas.

Question 36

In a randomized control trial of a vaccine, 255 out 1020 in the vaccine group subsequently developed the disease. in the placebo group, 352 out of 1100 developed the disease. Establish a 90% confidence interval estimate of the difference of proportions between the placebo and vaccine groups.

Write out the expression in the following form:

(A) (.25 - .32) ± 1.645*sqrt ( (.25*.75)/1020 +(.32*.68)/1100)

(B) (.25 - .32) ± 1.645*sqrt ( (.25*.75)/sqrt(1020) + (.32*.68)/sqrt(1100) )

(C) (.25 - .32) ± 1.96*sqrt ( (.25*.75)/1020 +(.32*.68)/1100)

(D) (.25 - .32) ± 1.96*sqrt ( (.25*.75)/sqrt(1020) + (.32*.68)/sqrt(1100) )

(E) (.25 - .32) ± 2.576*( sqrt((.25*.75))/1020 + sqrt((.32*.68))/1100)

Answer 36

A

1. We use the formula for difference in proportions (see formula below).
2. p_hat_1 = 255/1020, p_hat_2 = 352/1100
3. Critical z-scores for 90% are ± 1.645
4. Standard error is:

sqrt(

p_hat_1*(1-p_hat_1)/n_1

+

p_hat_2*(1-p_hat_2)/n_2))

Question 37

When making an inference about a population mean, which of the following suggests the use of z-scores rather than t-scores?

(A) Outliers are absent.

(B) The population is normal.

(C) The sample size is under 30.

(D) Strong skew is absent.

(E) The population standard deviation is known.

Answer 37

E

1. In the prescence of strong skewness or outliers, the t-procedures are contraindicated.
2. The t-procedures assume normality of the original population.
3. When the population standard deviation σ is unknown (the usual case) and the standard error s/sqrt(n) is substituted for σ/sqrt(n), then t-scores rather than z-scores are the proper choice.
4. In those rare cases where the population standard deviation is positively known, z-scores are appropriate.

Question 38

We have the following experimental data testing the efficacy of two herbs and a placebo on preventing the common cold.

H1 H2 Placebo

Sick 20 20 30

Not-sick 100 110 90

We have the following hypotheses:

Ho: Herbs do something, Ha: herbs do nothing.

What additional piece of information do we need before we can proceed?

Answer 38

Establish an alpha-level.

Question 39

We have the following experimental data testing the efficacy of two herbs and a placebo on preventing the common cold.

H1 H2 Placebo

Sick 20 20 30

Not-sick 100 110 90

We have the following hypotheses:

H_o: Herbs do something, H_a: herbs do nothing. &alpha=0.9

Answer 39

H1 H2 Placebo Total

Sick 20 20 30 80 (21%)

Exp-Sick 25.3 29.3 25.3

Not-sick 100 110 90 300 (79%)

Exp-Not-Sick 94.7

X² = (20-25.3)^2/25.3 + (30-29.4)^2/29.4 + (30-25.3)^2/25.3 + (100-94.7)^2/94.7 + (110-110.6)^2/110.6 + (90-94.7)^2/94.7 = 2.5287

Df = (r-1)*(c-1) = (2-1)*(3-1) = 2

For alpha = 10%, df == 2 the critical value is: 4.6

Since 2.53

Question 40

Formula for calculating the number of degrees of Freedom of a Contingency Table

Answer 40

(number_of_rows -1)*(number_of_columns-1)

Question 41

The acceptance rate at a particular college is 58%. If one takes a random sample of applicants to this college and constructs a confidence interval estimate of the acceptance rate, which of the following statements is true?

(A) The center of the interval would be 58%.

(B) The interval would contain 58%.

(C) A 99 percent confidence interval estimate would contain 58 percent.

(D) All of the above are true statements.

(E) None of the above are true statements.

Answer 41

E

There is no guarantee that the true population parameter will be within an interval based on a sample statistic.

Explain why.

Question 42

Which of the following are true?

(A) It is helpful to examine your data before deciding whether to use a one-sided or a two-sided hypothesis test.

(B) If the P-value is .05, the probability that the null hypothesis is correct is .05.

(C) The larger the P-value, the more evidence there is against the null hypothesis.

(D) If the P-value is small enough, we can conclude that the alternative hypothesis is true.

(E) We do not use sample statistics in stating the hypothesis.

Answer 42

E

1. The null and alternative hypotheses must be decided on before working on the data.
2. If the calculated P-value is small, we have evidence in support of the alternative hypothesis but this does not prove that the alternative hypothesis is true.
3. Hypotheses are about population parameters, not sample statistics.

Question 43

A theater owner believes that the attendance actually goes up if the best seats are advertised for higher prices. Regression analysis of Attendance (in 1000s) versus Price (in dollars) is shown below:

Dependent Variable is: Attendance

Source S sq df M Sq F

Regression 105.226 1 105.226 6.15

Residual 102.649 6 17.1002

Variable Coeff se t prob

Constant 19.2961 5.182 3.53 0.0124

Price 0.573437 0.2312 2.48 0.0470

R-sq = 40.6% R-sq (adj) = 42.4%

s= 4.136 df = 8-2 =6

What is the P-value for a t-test with Ho:β=0 and Ho β > 0 ?

(A) .0124

(B) .0239

(C) .0478

(D) .0956

(E) .5060

Answer 43

B

The computer output is for a two-sided test. Thus, in this case 2P = .0478 and P=.0239

Explan: what is beta? What do each of the terms in the output mean?

Question 44

Under what conditions would it be meaningful to construct a confidence interval estimate when the data consists of the entire population?

(A) If the population size is small ( n

(B) If the population size is large ( n ≥ 30)

(C) If a higher level of confidence is desired

(D) If the population is truly random

(E) Never

Answer 44

E

In determining confidence intervals, one uses sample statistics to estimate population parameters. If the data are actually the whole population, making an estimate has no meaning.

Ie, we do not have to make estimates; we can accurately characterize the data.

Question 45

The percentage of audiences for major news programs by broadcasters are as follows:

(ABC, NBC, CBS): 36 %

CNN: 42%

FOX: 22%

In a random sample of 50 viewers, 23 watched (ABC, NBC, CBS); 17 watched CNN and 10 watched FOX. If a goodness-of-fit test were performed, what would be the P-value?

Answer 45

X² = ∑ (obs-exp)²/exp

Expected values are: .36*50=18, .42*(50) = 21 and .22*(50) = 11.

𝜒² = (23-18)^2/18 + (17-21)^2/21 + (10-11)^2/11

Degrees of freedom are 3 -1 = 2

Question 46

A quality inspector plans to test a sample of 40 bottles of soft drink labeled as 20 ounces. If she finds the mean content to be less than 19.8 ounces, she will have the bottling machinery stopped an recalibrated. If it is known that the machine operates with a standard deviation of 0.75 ounces, what is the probability that the inspector will mistakenly stop a machine that is delivering a mean of at least 20 ounces per bottle?

(A) .023 (B) .046 (C) .050 (D) .092 (E) .119

Answer 46

B

We are being asked to calculate the probability of a Type 1 error.

σ_x = σ/sqrt(n) = 0.1186

The z-score of 19.8 is: 19.8-20/.1186 = -1.686

α = P (z ≤ - 1.686) = .046.

Notes:

1. We are given the population standard deviation here.
2. Not sure if alpha is used correctly here

Question 47

A ball machine turns out balls with diameter 8.55 mm. Each day a random sample of 5 balls is measured. If the mean diameter is under 8.35 mm or over 8.75 mm, the machinery is stopped and re-calibrated. This quality control procedure may be viewed as a hypothesis test with Ho: µ=8.55 and Ha: µ≠ 8.55. What would a Type II error result in?

(A) A warranted halt in production

(B) An unnecessary halt

(C) Continued production of wrong size balls

(D) Continued production of correctly sized balls

(E) Continued production of balls that are randomly the right or wrong size.

Answer 47

C

A Type II error is a mistaken failure to reject a false null hypothesis or, in this case, a failure to realize that the machine is turning out the wrong size ball.

Question 48

Define: Type II Error

Answer 48

A Type II error is a mistaken failure to reject a false null hypothesis

Question 49

Type I error

Answer 49

A type I error is the incorrect rejection of a true null hypothesis (a “false positive”), while a type II error is the failure to reject a false null hypothesis (a “false negative”).

More simply stated, a type I error is detecting an effect that is not present, while a type II error is failing to detect an effect that is present.

Question 50

The population mean µ=1250; Random sample is drawn with n=12, mean=1092, standard deviation = 308. What is the P-value?

Answer 50

P=0.0516

We use the t test.