Exam 3 Flashcards
define Statistical Inference
Statistical inference is the process by which we acquire information and draw conclusions about populations from samples.
The objective of estimation is to determine the _________ _______ of a population parameter on the basis of a sample statistic.
The objective of estimation is to determine the approximate value of a population parameter on the basis of a sample statistic.
Define Point Estimator
A point estimator draws inferences about a population by estimating the value of an unknown parameter using a single value or point.
Define Interval Estimator
An interval estimator draws inferences about a population by estimating the value of an unknown parameter using an interval.
An alternative statement is:
The mean income is between 380 and 420 $/week.
Define Unbiased estimator
An unbiased estimator of a population parameter is an estimator whose expected value is equal to that parameter.
An unbiased estimator is said to be _________ if the difference between the estimator and the parameter grows smaller as the sample size grows larger.
An unbiased estimator is said to be consistent if the difference between the estimator and the parameter grows smaller as the sample size grows larger.
If there are two unbiased estimators of a parameter, the one whose variance is smaller is said to be _______ ________.
If there are two unbiased estimators of a parameter, the one whose variance is smaller is said to be relatively efficient.
Interval Width : A ______ interval provides little information.
Interval Width: A wide interval provides little information.
Increasing the sample size _______ the width of the confidence interval while the confidence level can remain unchanged.
Increasing the sample size decreases the width of the confidence interval while the confidence level can remain unchanged.
Error of estimation
We can define the sampling error as the difference between an estimator and a parameter. Specifically referred to as error of estimation
What is B mean and stand for in the sample size for proportion equation?
B in sample size for proportion equation is the maximum error of estimation that we are willing to tolerate, B stands for bound in the error of estimation.
The objective of estimation is to determine the approximate value of a population parameter on the basis of a ______ _________
The objective of estimation is to determine the approximate value of a population parameter on the basis of a sample statistic
That is we say (with some ___% certainty) that the __________ _________ of interest is between some lower and upper bounds.
That is we say (with some ___% certainty) that the population parameter of interest is between some lower and upper bounds.
The objective of estimation is to determine the approximate value of a _________ _________ on the basis of a sample statistic
The objective of estimation is to determine the approximate value of a population parameter on the basis of a sample statistic
what is Confidence Interval
It states that there is 1 - α probability that the sample mean will be equal to a value such that the interval
p(hat) plus or minus z alpha/2 sq rt ( p(hat) (1-p(hat)) / n )
will include the population proportion. Once the sample proportion is computed, the interval acts as the lower and upper limits of the interval estimate of the population proportion.
The probability 1 – α is the ________ ______, which is a measure of how frequently the interval will actually include p or µ.
The probability 1 – α is the confidence level, which is a measure of how frequently the interval will actually include p or µ.
Fail to reject null = ?
Fail to reject null = agreed with null hypothesis
Reject null = ?
Reject null = agreed with alternative hypothesis
What equation should you use:
Determine the sample size to estimate a population proportion within .02 with 99% confidence with p = .65
n = ( z α/2 sqrt(p(1-p)) / B ) ^2
Sample size for proportion Chapter 9
What equation should you use:
A marketing manager wanted to know whether his new product design was going to be popular. He asked a random sample of 500 potential customers whether they liked the new design better than the current design. A total of 320 people said that they did. Estimate with a 95% confidence the proportion of all potential customers who would prefer the new design.
P(hat)= x / n
p(hat) plus or minus z alpha/2 sq rt ( p(hat) (1-p(hat)) / n )
Sample proportion and confidence interval for the proportion
Chapter 9
A ____ ___ ____ occurs when we reject a true null hypothesis. That is, a ____ ___ ____ occurs when the jury convicts an innocent person.
A Type I error occurs when we reject a true null hypothesis. That is, a Type I error occurs when the jury convicts an innocent person.
A ____ ___ ____ occurs when we don’t reject a false null hypothesis. That is, a ____ ___ ____ occurs when a guilty defendant is acquitted, let go.
A Type II error occurs when we don’t reject a false null hypothesis. That occurs when a guilty defendant is acquitted, let go.
What is a hypothesis
A hypothesis is a claim (assertion) about
a population parameter:
population mean
Ex: The mean monthly cell bill in city is µ = $42
population proportion
Ex: The proportion of adults in this city with cells is p = 0.68
H0: — the ‘null’ hypothesis always uses what symbols
equals, greater or equal to, less than or equal to
“=” , “≤” or “≥” sign
H1: — the ‘________’ or ‘________’ hypothesis
H1: — the ‘alternative’ or ‘research’ hypothesis
What is the alternative Hypothesis?
Is the opposite of the null hypothesis
e.g., The average number of TV sets in U.S. homes is not equal to 3 ( H1: μ ≠ 3 )
Challenges the status quo
Never contains the “=” , “≤” or “≥” sign
May or may not be proven
Is generally the hypothesis that the researcher is trying to prove
P(Type I error) = ?
P(Type I error) = α
P(Type II error) = ?
P(Type II error) = β
two possible errors
Type I error: Reject a true null hypothesis
Type II error: Do not reject a false null hypothesis.
α is called the ___________ _____
Set by researcher in advance
α is called the significance level
Set by researcher in advance
What are critical values
Critical values are what define the rejection region, There are two cutoff values (critical values), defining the regions of rejection
Two-tail test
rejection region is in both tails, non directional, α / 2
what are the six-steps to hypothesis testing
State the appropriate null and alternative hypotheses
H0: μ = 3 H1: μ ≠ 3 (This is a two-tail test)
2. Specify the desired level of significance and the sample size
Suppose that α = 0.05 and n = 100 are chosen for this test
3. Determine the critical values
For α = 0.05 the critical Z values are ±1.96
4. Collect the data and compute the test statistic
Suppose the sample results are
n = 100, X-bar = 2.84 (σ = 0.8 is assumed known) [Zstat Equation]
5. Is the test statistic in the rejection region? [Graph]
6. Reach a decision and interpret the result
What is P-value hypothesis testing
p-value: Probability of obtaining a test statistic equal to or more extreme than the observed sample value given H0 is true
The p-value is also called the observed level of significance
It is the smallest value of α for which H0 can be rejected
If the p-value is ___ then H0 must go
low
_________________ H0 since p-value = 0.0136 < α = 0.05
Reject H0 since p-value = 0.0136 < α = 0.05
Use which equation to find test statistic
Zstat
Which equation should you use:
Has the recent drop in airplane passengers resulted in better on-time performance? Before the recent downturn one airline bragged that 92% of its flights were on-time. Can we conclude at the 5% significance level that the airline’s on-time performance has improved?
p(hat) and Zstat
Zstat = (p(hat) - p) / (sq rt(p(1-p)/n))
Chapter 10 One-Tailed
Critical values you _____
critical values you lookup
test statistics you ______
Test statistics you calculate
Which equations should you use: The university of Memphis advertises that its average class size is 35 students or less. A student organization is concerned that budget cuts have let to increased class sizes and would like to test this claim. A sample of 38 classes were selected and the average class size is 36.9 students. The standard deviation for the entire college is 8 students. Using an alpha = .05, does the student have enough evidence to show that class sizes have increased?
Six Step Method, One-Tailed, Zstat Sigma known
What is degrees of freedom
Number of observations that are free to vary after sample mean has been calculated (ν = n – 1)
Degrees of freedom = ?
Degrees of freedom = n - 1
tstat < - tcrit or tstat > tcrit
Reject
Interested in a population’s central location: It has a Z-Distribution so we use a Z-test statistic when the Sigma Squared is ________
Interested in a population’s central location: It has a Z-Distribution so we use a Z-test statistic when the Sigma Squared is known
Interested in a population’s central location: It has a t-Distribution so we use a t-test statistic when the Sigma Squared is ________
Interested in a population’s central location: It has a t-Distribution so we use a t-test statistic when the Sigma Squared is unknown
How to read the t-table
(over) Upper tail area = alpha/2
(down) degrees of freedom = n-1
If we are interested in drawing inferences about a population’s variability, we investigate population variance: σ^2 using ______ _________
If we are interested in drawing inferences about a population’s variability, we investigate population variance: σ^2 using sample variance (s^2)
If we are interested in drawing inferences about a population’s variability, It has a chi-squared distribution so we use a ___ _______ ________, with n–1 degrees of freedom.
If we are interested in drawing inferences about a population’s variability, It has a chi-squared distribution so we use a χ2 test statistic, with n–1 degrees of freedom.
when comparing two populations, we examined ___________ __________
when comparing two populations, we examined independent samples.
If, however, an observation in one sample is matched with an observation in a second sample, this is called a ________ _____ __________.
If, however, an observation in one sample is matched with an observation in a second sample, this is called a matched pairs experiment.
what is the F test used for
to compare two population variances
What is the equation for testing a population means, with equal variances?
t = ( (xbar1-xbar2) - (mu1-mu2) ) / ( sqrt(Sp^2(1/n1+1/n2) )
What is the degree of freedom for an F test?
V1 = n1-1 and V2 = n2-1
What is the equation for pooled variance
Sp^2 = ( (n-1) s1^2 + (n2-1) s2^2 ) / ( n1+n2-2 )
How to read the F Table
numerator degrees of freedom determine the column
denominator degrees of freedom determine the row
T/F? If a sample has 25 observations and we need a 95% confidence estimate for the population mean with an unknown standard deviation, the appropriate value of t is 2.064.
true
Which of the following conditions is needed regarding the chi-squared test statistic for the test of variance?
Question options:
a) The population random variable must be normal.
b) The test statistic must be a non-negative number.
c) The test statistic must have a chi-squared distribution with n - 1 degrees of freedom.
d) All of these choices are true.
d) all of these choices are true
Researchers determined that 60 Puffs tissues is the average number of tissues used during a cold. Suppose a random sample of 96 persons yielded the following data on the number of tissues used during a cold: sample mean x-bar= 52 and s = 22. Suppose the alternative we wanted to test was H1: m 1.661
b) Reject H0, if t 1.985 or Z
b) Reject H0, if t
T/F? the lower limit of 90% confidence interval for the population proportion p, given that n = 400 and the sample proportion p-hat=.10, is .0247
False
T/F? Increasing the probability of a Type I error will increase the probability of a Type II error
False
T/F? The statement of the null hypothesis always includes an equals sign
True
What would be an appropriate alternative hypothesis?
The population mean is not equal to 40
INCORRECT:
The sample mean is equal to 40
The population mean is equal to 40
The sample mean is not equal to 40
T/F? Reject H0, when Zstat < - Zcritical
True
T/F? Reject H0, when Zstat < Zcritical
False
T/F? Reject H0, when Zstat > - Zcritical
False
T/F? Reject H1, when Zstat < - Zcritical
False
If the p-value is < alpha, then
a) you fail to reject the null hypothesis
b) reject the null hypothesis
if the p-value is low, it must go.
b
