Deck 12: Hypothesis Tests Applied to Means: One Sample

Question 1

In one-sample tests of means we

A) compare one sample mean with another.
B) compare one sample mean against a population mean.
C) compare two sample means with each other.
D) compare a set of population means.

Answer 1

compare one sample mean against a population mean.

Question 2

An assumption behind the use of a one-sample t test is that

A) the population is normally distributed.
B) the sample is normally distributed.
C) the population variance is normally distributed.
D) the population variance is known.

Answer 2

the population is normally distributed.

Question 3

The standard error of the mean is

A) equal to the standard deviation of the population.
B) larger than the standard deviation of the population.
C) the standard deviation of the sampling distribution of the mean.
D) none of the above

Answer 3

the standard deviation of the sampling distribution of the mean.

Question 4

If the population from which we sample is normal, the sampling distribution of the mean

A) will approach normal for large sample sizes.
B) will be slightly positively skewed.
C) will be normal.
D) will be normal only for small samples.

Question 5

When we are using a two-tailed hypothesis test, the alternative hypothesis is of the form

A) H ₁ : µ ≠ 50.
B) H ₁ : µ
C) H ₁ : µ > 50.
D) H ₀ : µ = 50.

Question 6

When you are using a one-sample t test, the degrees of freedom are

A) N.
B) N - 1.
C) N + 1.
D) N - 2.

Question 7

I want to test the hypothesis that children who experience daycare before the age of 3 do better in school than those who do not experience daycare. I have just described the

A) alternative hypothesis.
B) research hypothesis.
C) experimental hypothesis.
D) all of the above

Question 8

The standard error of the mean is a function of

A) the number of samples.
B) the size of the samples.
C) the standard deviation of the population.
D) both b and c

Question 9

It makes a difference whether or not we know the population variance because

A) we cannot deal with situations in which the population variance is not known.
B) we have to call the result t if the population variance is used.
C) we have to call the result z if the population variance is not used.
D) we have to call the result t if the sample variance is used.

Question 10

Suppose that we know that the sample mean is 18 and the population standard deviation is 3. We want to test the null hypothesis that the population mean is 20. In this situation we would

A) reject the null hypothesis at a = .05.
B) reject the null hypothesis at a = .01
C) retain the null hypothesis.
D) We cannot solve this problem without knowing the sample size.

Question 11

Which of the following is NOT part of the Central Limit Theorem?

A) The mean of the sampling distribution approaches the population mean.
B) The variance of the sampling distribution approaches the population variance divided by the sample size.
C) The sampling distribution will approach a normal distribution as the sample size increases.
D) All of the above are part of the Central Limit Theorem.

Question 12

The importance of the underlying assumption of normality behind a one-sample means test

A) depends on how fussy you are.
B) depends on the sample size.
C) depends on whether you are solving for t or z .
D) doesn't depend on anything.

Question 13

In using a z test for testing a sample mean against a hypothesized population mean, the formula for z is

A)
B)
C)
D) none of the above

Question 14

If the population from which we draw samples is "rectangular," then the sampling distribution of the mean will be

A) rectangular.
B) normal.
C) bimodal.
D) more normal than the population.

Question 15

If we knew the population mean and variance, we would expect

A) the sample mean would closely approximate the population mean.
B) the sample mean would differ from the population mean by no less than 1.96 standard deviations only 5% of the time.
C) the sample mean would differ from the population mean by no more than 1.64 standard deviations only 5% of the time.
D) the sample mean would differ from the population mean by more than 1.96 standard errors only 5% of the time.

Question 16

When we are using a two-tailed hypothesis test, the null hypothesis is of the form

A) H ₁ : µ ≠  50.
B) H ₁ : µ
C) H ₁ : µ > 50.
D) H ₀ : µ = 50.

Question 17

The sampling distribution of the mean is

A) the population mean.
B) the distribution of the population mean over many populations.
C) the distribution of sample means over repeated samples.
D) the mean of the distribution of the sample.

Question 18

With large samples and a small population variance, the sample means usually

A) will be close to the population mean.
B) will slightly underestimate the population mean.
C) will slightly overestimate the population mean.
D) will equal the population mean.

Question 19

Many textbooks (though not this one) advocate testing the mean of a sample against a hypothesized population mean by using z even if the population standard deviation is not known, so long as the sample size exceeds 30. Those books recommend this because

A) they don't know any better.
B) there are not tables for t for more than 30 degrees of freedom.
C) the difference between t and z is small for that many cases.
D) t and z are exactly the same for that many cases.

Question 20

If the standard deviation of the population is 15 and we repeatedly draw samples of 25 observations each, the resulting sample means will have a standard error of

A) 2
B) 3
C) 15
D) 0.60

Question 21

If we have calculated a confidence interval and we find that it does NOT include the population mean,

A) we must have done something wrong in collecting data.
B) our interval was too wide.
C) we made a mistake in calculation.
D) this will happen a fixed percentage of the time.

Question 22

Cohen's
is an example of

A) a measure of correlation.
B) an r -family measure.
C) a d -family measure.
D) a correlational measure.

Question 23

The reason why we need to solve for t instead of z in some situations relates to

A) the sampling distribution of the mean.
B) the sampling distribution of the sample size.
C) the sampling distribution of the variance.
D) the size of our sample mean.

Question 24

A one-sample t test was used to see if a college ski team skied faster than the population of skiers at a popular ski resort. The resulting statistic was t _.05(23) = -7.13, p

A) The sample mean of the college skiers was significantly different from the population mean.
B) The sample mean of the college skiers was not significantly different from the population mean.
C) The null hypothesis was true.
D) The sample mean was greater than the population mean.

Question 25

If we have run a t test with 35 observations and have found a t of 3.60, which is significant at the .05 level, we would write

A) t (35) = 3.60, p <.05
B) t (34) = 3.60, p >.05.
C) t (34) = 3.60, p <.05
D) t (35) = 3.60, p >.05

Question 26

If we fail to reject the null hypothesis in a t test we can conclude

A) that the null hypothesis is false.
B) that the null hypothesis is true.
C) that the alternative hypothesis is false.
D) that we don't have enough evidence to reject the null hypothesis.

Question 27

Which of the following statements is true?

A) Confidence intervals are the boundaries of confidence limits.
B) Confidence intervals always enclose the population mean.
C) Sample size does not affect the calculation of t .
D) Confidence limits are the boundaries of confidence intervals.

Question 28

When we take a single sample mean as an estimate of the value of a population mean, we have

A) a point estimate.
B) an interval estimate.
C) a population estimate.
D) a parameter.

Question 29

All of the following increase the magnitude of the t statistic and/or the likelihood of rejecting H₀ EXCEPT

A) a greater difference between the sample mean and the population mean.
B) an increase in sample size.
C) a decrease in sample variance.
D) a smaller significance level ( a ).

Question 30

A t test is most often used to

A) compare two means.
B) compare the standard deviations of two samples.
C) compare many means.
D) none of the above

Question 31

A 95% confidence interval is going to be _______ a 99% confidence interval.

A) narrower than
B) wider than
C) the same width as
D) more accurate than

Question 32

The variance of an individual sample is more likely than not to be

A) larger than the corresponding population variance.
B) smaller than the corresponding population variance.
C) the same as the population variance.
D) less than the population mean.

Question 33

Which of the following statistics comparing a sample mean to a population mean is most likely to be significant if you used a two-tailed test?

A) t = 10.6
B) t = 0.9
C) t = -10.6
D) both a and c

Question 34

The two-tailed p value that a statistical program produces refers to

A) the value of t .
B) the probability of getting at least that large a value of t if the null hypothesis is false.
C) the probability of getting at least that large an absolute value of t if the null hypothesis is true.
D) the probability that the null hypothesis is true.

Question 35

For a t test with one sample we

A) lose one degree of freedom because we have a sample.
B) lose one degree of freedom because we estimate the population mean.
C) lose two degrees of freedom because of the mean and the standard deviation.
D) have N degrees of freedom.

Question 36

Which of the following does NOT directly affect the magnitude of t ?

A) The actual obtained difference ( - µ).
B) The magnitude of the sample variance ( s ²).
C) The sample size ( N ).
D) The population variance (σ²).

Question 37

With a one-sample t test, the value of t is

A) always positive.
B) positive if the sample mean is too small.
C) negative whenever the sample standard deviation is negative.
D) positive if the sample mean is larger than the hypothesized population mean.

Question 38

If we compute 95% confidence limits on the mean as 112.5 - 118.4, we can conclude that

A) the probability is .95 that the sample mean lies between 112.5 and 118.4.
B) the probability is .05 that the population mean lies between 112.5 and 118.4.
C) an interval computed in this way has a probability of .95 of bracketing the population mean.
D) the population mean is not less than 112.5.

Question 39

When are we most likely to expect larger differences between group means?

A) when there is considerable variability within groups
B) when there is very little variability within groups
C) when we have large samples
D) when we have a lot of power

Question 40

The sampling distribution of the variance is

A) positively skewed.
B) negatively skewed.
C) normal.
D) rectangular.

Question 41

The standard deviation of a sampling distribution is known as the standard error.

Question 42

The larger the difference between the sample mean and the population mean, the larger the t value.

Question 43

Assuming a two-tailed one sample test is being used, what are the critical values for t given the following sample sizes:
a. N = 10
b. N = 15
c. N = 30

Question 44

According to the Central Limit Theorem, as the number of samples increases, the distribution will approach the normal distribution.

Question 45

t-tests are used to test a sample mean when the population mean is unknown.

Question 46

When the population standard deviation is known, z scores are appropriate to test a sample mean.

Question 47

A normal distribution is one in which all outcomes are equally likely.

Question 48

A confidence interval computed for the mean of a single sample

A) defines clearly where the population mean falls.
B) is not as good as a test of some hypothesis.
C) does not help us decide if there is a significant effect.
D) is associated with a probability statement about the location of a population mean.

Question 49

A sampling distribution of the mean is typically the mean of one sample.

Question 50

When comparing the mean of a sample of 30 people to the population mean, the degrees of freedom are 31.

Question 51

The confidence intervals for two separate samples would be expected to differ because

A) the sample means differ.
B) the sample standard deviations differ.
C) the sample sizes differ.
D) all of the above

Question 52

The point of calculating effect size measures is to

A) decide if something is statistically significant.
B) convey useful information to the reader about what you found.
C) reject the null hypothesis.
D) prove causality.

Question 53

When you have a single sample and want to compute an effect size measure, the most appropriate denominator is

A) the variance of the sample.
B) the standard deviation of the sample.
C) the sample size.
D) none of the above

Question 54

The t distribution

A) is smoother than the normal distribution.
B) is quite different from the normal distribution.
C) approaches the normal distribution as its degrees of freedom increase.
D) is necessary when we know the population standard deviation.

Question 55

Given  m = 100,
s = 27, and N = 30:
a. Calculate t.
b. Write a sentence interpreting the value of t as a two-tailed test.
c. Write a sentence interpreting the value of t as a one-tailed test.

Question 56

When would you NOT use a standardized measure of effect size?

A) when the difference in means is itself meaningful
B) when it is clearer to the reader to talk about a percentage
C) when some other measure conveys more useful information
D) all of the above

Question 57

Given a sample size of 30, and one sample t = -2.5, what would you conclude about the sample from which the mean was drawn?

Question 58

The term "effect size" refers to

A) how large the resulting t statistic is.
B) the size of the p value, or probability associated with that t .
C) the actual magnitude of the mean or difference between means.
D) the value of the null hypothesis.

Question 59

Student's t distribution essentially accounts for the fact that t is often larger than the corresponding z because it is based on estimated variance, which is biased.

Question 60

As the sample variability increases, the magnitude of t increases.

Question 61

Calculate the 95% confidence interval for μ given
s = 25, and N = 101.

Question 62

Briefly describe two factors that affect the magnitude of t .

Question 63

In the previous question, what would be the minimum mean score of the teacher's students that would yield a statistically significant difference using a one-tailed test?

Question 64

The following 10 numbers were drawn from a population.
            5           7           7           10         10         10         11         12         12         13
a. Calculate the 95% confidence interval for the population mean.
b. Is it likely that these numbers came from a population with a mean of 13? Explain.

Question 65

The average SAT score for a local high school was 1100. One teacher is convinced that the 25 students who were in his homeroom performed better than the average student in the high school. Their average score was 1125 with a standard deviation of 100.
a. Calculate t .
b. Evaluate the teacher's "hypothesis" in light of t .

Question 66

The mean anxiety score in elementary school children is 14.55. A researcher wants to know if children of anxious parents are more anxious than the average child. Below are the anxiety scores from 10 children of anxious parents.
13         14         14         15         15         15         16         17         17         18
 
a. Calculate the t value.
b. Write a sentence to answer the researcher's question.

Question 67

Explain the following statement: p (100 ≤ μ ≤ 110) = .90.