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Stats > Reading Quiz 10 > Flashcards

Reading Quiz 10 Flashcards

1
Q

level C confidence interval for parameter parts

A

confidence interval

confidence level C

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2
Q

confidence interval

A

calculated from data
usually estimate ± margin of error
best guess for value of unknown parameter

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3
Q

margin of error

A

m ±

shows how accurate believe guess is, based on the variability of the estimate

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4
Q

confidence level C

A

gives probability that the interval will capture the true parameter value in repeated samples
success rate for method

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5
Q

one sample z interval for means

A

σ known

chooses SRS of size n for population having unknown mean μ and known standard deviation σ

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6
Q

level C confidence interval for μ when one sample z interval for means

A

x̅ ± z*(σ/√n)

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7
Q

critical value z*

A

value that determines area C between -z* and z* under the standard Normal curve
interval is exact when population distribution is Normal and is approximately correct for large n in other cases (because of central limit theorem)

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8
Q

sample size for desired margin of error

A

to determine sample size n that will yield a confidence interval for a population mean with a specific margin of error m, set expression for margin of error to be less than or equal to and solve for n:
z*(σ/√n) ≤ m
always round up to the next whole number when finding n

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9
Q

margin of error of confidence interval gets smaller when

A

confidence level C decreases (z* gets smaller)
population standard deviation σ decreases
sample size n increases

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10
Q

steps to construct a confidence interval

A
  1. parameter
  2. conditions
  3. calculations
  4. interpretation
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11
Q

parameter

A

identify the population of interest and the parameter you want to draw conclusions about
state the C-level

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12
Q

conditions

A

state the name of the appropriate inference procedure
verify the conditions for using if
if assuming that any conditions are met then clearly say so
if any conditions are clearly not met then state that they are not met and you will “proceed with caution”

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13
Q

calculations

A

carry out the inference procedure

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14
Q

interpretation

A

interpret results in context of problem

conclusion, connection, context

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15
Q

one sample t interval for means (σ unknown)

A

in practice, don’t know σ
exact level C confidence interval for mean μ of a Normal population with unknown σ is
x̅ ± t(s/√n)
t is critical value of t distribution with n-1 degrees of freedom

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16
Q

degrees of freedom

A

n-1

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17
Q

one sample t interval exactly correct when

A

population distribution is Normal and is approximately correct for large n in other cases

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18
Q

standard error

A

when standard deviation of statistic is estimated from data, result is standard error of statistic
standard error of sample mean x̅ is (s/√n)

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19
Q

to compare responses to two treatments in matched pairs or before and after measurements on same subjects

A

apply one sample t procedures to observed difference

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20
Q

conditions for inference about a population mean

A

SRS
normality
independence

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21
Q

SRS condition

A

data are SRS of size n from population of interest or SRS from randomized experiment

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22
Q

normality condition

A

if population is normal then so is sampling distribution of x bar
if population is not normal but n is large (greater than or equal to 30) then sampling distribution is approximately normal according to central limit theorem
if population is not normal and if n is small, observe shape of the sample data: it is enough that the distribution be roughly symmetric and unimodal to make the t interval reliable

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23
Q

independence condition

A

assume that individual observations are independent

when sampling without replacement must verify that population of size N is at least 10 times the sample size

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24
Q

one sample z interval for proportions

A

level C confidence interval for p is
p̂ ± z* (√(p̂ˆq/n)
z* is the upper (1-C)/2 critical value from the standard Normal distribution

25
Q

conditions for inference about a population proportion

A

SRS: same as SRS condition for inference about a population mean
Normality: sample is large enough to satisfy counts of success np̂ and failures nˆq are both greater than or equal to 10
independence: same as the independence condition for inference about a population mean

26
Q

reminder

A

central limit theorem cannot be applied to proportions

27
Q

sample size needed to obtain a confidence interval with approximate margin of error m for a population proportion involves solving

A

z(√(pq/n)) less than or equal to m for n
p is a guessed value for the sample proportion p̂
z* is the critical value for the level of confidence you want

28
Q

trick for z(√(pq*/n)) formula

A

if you use p*=0.5 in this formula, the margin of error of the interval will be less than or equal to m no matter what the value of p̂ is

29
Q

Statistical inference provides methods for drawing conclusions about a ____ from ____.

A

A. population, sample data

30
Q

Suppose we know that the sd of the sample mean (a.k.a. standard error of the mean) is 4.5. This implies that if we were to draw many samples from the population, about 95% of these sample means would fall within what interval?

A

A. The population mean plus or minus 9.

31
Q

True or False: we should imagine the sample mean as being at the center of a bell-shaped curve, with 2 standard deviations of the sample mean (a.k.a. standard errors) on either side of this point encompassing 95% of the other sample means.(Assume the sample is an SRS and the sample means are normally distributed.)

A

A. False. We should imagine the population mean as being at the center of that bell curve. We visualize the sample mean as falling within 2 standard errors of the population mean 95% of the time.

32
Q

True or False: The reasoning we use in making confidence intervals around a sample mean is as follows: if the sample mean is normally distributed, then 95% of the time, x-bar will be within 2 sample standard deviations (standard errors) of the population mean, mu. Whenever x-bar is within 2 standard errors of mu, mu is within 2 standard errors of x-bar. So if we make an interval + or – 2 standard errors around x-bar, that interval will encompass mu for 95% of the sample means we obtain.

A

true

33
Q

Someone says, “I read that the 95% confidence interval for a certain group’s score on a certain test was 115 to 128. That means that 95% of all the members of the group score in that range.” Is this an accurate interpretation? If not, please give a better one.

A

A. Not accurate. The confidence interval stated means that we are 95% confident that the population mean lies within the stated interval. And 95% confident means that 95% of the intervals obtained the way we got this one would encompass the population mean.

34
Q

In order to construct a confidence interval for a mean, what conditions need to be met?

A

A. That the data come from a SRS of the population, that the sampling distribution of the x-bar is
approximately normal, and that the individual observations are independent.

35
Q

A first person says, “I want a 90% confidence interval. So I’ll look in the normal table for the z-score with 95% of the area to the left of it.” A second person says, “You mean 90% of the area, don’t you?” What is the correct way to look in the table?

A

A. The first person got it right. The region around the population mean that subsumes 90% of the sample means is that with 5% above that region and 5% below that region. So you want z for .95 or the negative of the z for .05.

36
Q

What does the symbol z* stand for?

A

A. The z-score with (1-C)/2 of the area lying to the right of it. Or: the number of standard deviations
above and below the mean that bound the C level confidence interval.

37
Q

True or False: The values mu - z* (σ/√n) and mu + z* (σ/√n) represent the upper and lower bounds for the nn confidence interval for the mean.

A

A. False. The confidence interval is centered around x-bar, not around mu, because we don’t know mu. (If we did, we wouldn’t need to make a confidence interval.) The values listed above are the bounds within which there is a probability C that any observed sample mean will fall.

38
Q

If my friend’s age falls in the interval of my age plus or minus 5 years, then my age must fall within the interval of my friend’s age plus or minus 5 years. Is this true, and is this sort of reasoning central to the reasoning about confidence intervals?

A

yes and yes

39
Q

True or False: The way in which the statement in the previous question has its analogy in the reasoning about confidence intervals is: any time the sample mean falls within the interval of mu plus or minus the margin of error, then the population mean must fall within the interval of x-bar plus or minus the same margin of error.

A

true

40
Q

Example 10.5 on page 630 is worthy of careful study. What are the 4 steps that were exemplified in using confidence intervals?

A

A. 1. Identify the population of interest and the parameter to be estimated. 2. State the appropriate procedure, and verify that the conditions for using it are met. 3. Carry out the procedure. CI = estimate ± margin of error. 4. Interpret the results in the context of the problem.

41
Q

Please tell whether the margin of error (which is half the width of the confidence interval), or the width of the confidence interval itself, gets bigger or smaller under each of the following circumstances: a. the population standard deviation gets smaller, b. the level of confidence C gets bigger (e.g. a move from a 90% confidence interval to a 99% confidence interval) c. the sample size gets bigger, and d. the population size gets bigger?

A

A. a. smaller, b. bigger, c. smaller, d. no effect

42
Q

Is it preferable in research for a 95% confidence interval to have its upper and lower bounds closer together, or farther apart?

A

A. Closer together, because this represents a more accurate estimate of whatever you’re trying to estimate.

43
Q

Suppose you are a researcher planning a study, and you are deciding how many subjects to enroll. You want a certain margin of error m. You know what level of confidence you want, and you know (or estimate) the sigma for the population. How do you figure out the sample size?

A

A. Set z* (σ/√n) less than or equal to m and solve that inequality for n. As usual, you use as z* the z score that has (1-C)/2 n area to the right of that score.

44
Q

What two conditions does our text list for inference about means when the population standard deviation is not known?

A

A. That the data are a SRS from the population of interest, and that the observations from the population have a normal distribution.

45
Q

The sample standard deviation divided by the square root of n is called the ______ ______ of the sample mean.

A

standard error

46
Q

When the standard deviation of any statistic is estimated from the data, the result is called the ____ ____ of that statistic. (Thus you can have these that apply not just to the sample mean.)

A

standard error

47
Q

Does it make sense to speak of the standard deviation of the population mean? If not, why not?

A

A. A population parameter is a single number, not a random variable. As such, it doesn’t have a variance
or standard deviation. σ/√n gives the standard deviation of the sample mean, and s/√n gives a less accurate estimate of the standard deviation of the sample mean.

48
Q

The z-statistic is (x̅-μ)/(σ/√n). What is the t-statistic?

A

A. (x̅-μ)/(s/√n) where s is the sample standard deviation

49
Q

There is just one standard normal distribution. Is there just one t-distribution?

A

A. No; there is a t-distribution for each number of degrees of freedom of the statistic, where the degrees
of freedom in dealing with means is n-1.

50
Q

What is the general shape of the t-distribution?

A

A. Bell-shaped, similar to the normal.

51
Q

As the degrees of freedom increase, the shape of the t-distribution more and more closely approximates what?

A

A. The standard normal distribution.

52
Q

Can you please explain the reason for the way the shape of the t-distribution differs from that of the normal when the degrees of freedom are low?

A

A. The distribution is more spread out, less peaked, with less probability in the center and more in the tails – in other words, it has more variation. This is because estimating sigma by s rather than knowing sigma for sure adds more variation to the statistic.

53
Q

What’s the expression for the level C confidence interval for the population mean (mu), using the t distribution to estimate when the population standard deviation is unknown?

A

A. The confidence interval is x̅ ± t(s/√n)
where t is the upper (1-C) critical value for the t distribution with n-1 degrees of freedom. (And s is the sample standard deviation, and n is the sample size.)

54
Q

The statistic that estimates (in an unbiased way) the population proportion is ____.

A

A. The sample proportion.

55
Q

What is the standard deviation of the sample proportion (provided the population is at least 10 times as big as the sample)?

A

A. √(pq/n) where p is the population proportion, q is 1-p, and n is the sample size.

56
Q

If np and nq are at least 10, then we can treat the distribution of p-hat as approximately what?

A

normal

57
Q

True or False: other than the method described in the problem above, the Central Limit Theorem can also be applied to a sample of proportions to establish Normality.

A

A. False. The Central Limit Theorem can only be applied to sample means not proportions.

58
Q

In order to determine the sample size needed to obtain a confidence interval with approximate margin of
error m for a population proportion you must solve z(√(pq/n)) less than or equal to m for n, where p is a guessed value for the sample proportion pˆ . For p*, if you are not given a reasonable guessed value you should substitute _____ in the formula.

A

A. 0.5. This will make the margin of error of the interval will be less than or equal to m no matter what the value of pˆ is.

59
Q

What is the expression for a confidence interval around the sample proportion?

A

A. The confidence interval is pˆ ± z(√(ˆpˆq/n)) . This fits the format of estimate ± z SEestimate for any n
normally distributed estimator.

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