Deck 12: Introduction to Statistical Inference: Testing Hypotheses About Single Means Z and T

Question 1

The human resources director of a large corporation determines the typing speeds of a random sample of secretaries from his company on certain standard materials. He wishes to test the hypothesis that the mean for his population is equal to 50 words per minute, the national norm for secretaries on these materials. Explain in general terms the logic and procedures for testing his hypothesis.

Answer 1

Compare

with sample means expected through random sampling if hyp:

were true; if

typical, retain hypothesis as a reasonable possibility; if

were very atypical, reject hypothesis as unreasonable.

Question 2

Compute the best estimates of

and

for each of the following samples:?
(a)pulse rate: 72, 86, 75, 66, 90;?
(b)number of tumors per rat: 2, 7, 8, 6, 6, 11, 3.

Answer 2

(a)

(b)

Question 3

In the following problem, locate the regions of rejection and the sample results on a rough distribution sketch; perform the test; and give final conclusions about the value of


sample: 15, 13, 12, 8, 15, 12.

Answer 3

;
retain
we cannot conclude anything.

Question 4

The Director of Student Health Services wishes to know if her anti-smoking campaign has been effective. Last year, a complete survey of all students found the mean number of cigarettes smoked per day by students to be 8.6 with a standard deviation of 2.0. The Director of the Student Health Service obtains the following results from a recently selected random sample of students (in cigarettes smoked per day):

(a)Set up

and

(b)Perform the statistical test (

(c)Draw final conclusions.

Question 5

A large suburban school district is contemplating instituting a special program to combat overweight in elementary school boys. In order to determine whether such a program is warranted, a random sample of 36 fourth-grade boys is selected. Then, for each boy the difference between his actual weight and the weight recommended for his height and age is recorded. A positive difference indicates overweight and a negative difference, underweight. The following results are obtained (in pounds):


(a) Set up

and

.
(b)Perform the statistical test ( (c)Draw final conclusions.

Question 6

Glo-lite light bulbs are advertised as having an average life of 600 hours with a standard deviation of 20. Consumer's Advocates (CA) has received numerous complaints that the bulbs burn out much more quickly. In order to check on the validity of these complaints, CA tests a sample of 25 bulbs and obtains:

(a)Set up

and

(b)Perform the statistical test (

(c)Draw final conclusions.?(d)?Now repeat using

Question 7

Ms. Phillips, research director of a large city school district, wishes to compare the mean score on a widely used reading readiness test of entering first graders in her district with the national norm of 62 and a standard deviation of 8. The results for a random sample of 10 children from her district are:

(a) Set up

and

(b)Perform the statistical test (

(c)Draw final conclusions.
(d)Now repeat using

Question 8

A training director for a large company has been told that on completion of the training course he has been using for some time, the average score of his trainees should be at least 100. His only concern is whether he will have to begin remedial steps to insure that the population of trainees is not below standard. A sample of 15 recent trainees finds:

(a) Set up

and

(b)Perform the statistical test (

(c)Draw final conclusions.
(d)Now repeat using

Question 9

The task in a particular concept-formation experiment is to discover, through trial and error, the correct sequence in which to press a row of buttons. It is determined from the nature of the task that the average score obtained by random guessing alone would be 20 correct out of a standard series of trials. The following are the scores for a sample of volunteer college students: 31, 24, 21, 25, 32. You wish to determine whether such subjects do better, on the average, than expected by just guessing.(a)Set up

and

(b)Determine the critical value(s) of

(c) Perform the statistical test.?
(d) Draw final conclusions.

Question 10

The new Streamer four-cylinder compact is advertised as obtaining 39.2 miles per gallon (estimated highway mileage). A competitor doubts their claim and tests a sample of Streamers under highway conditions with the following results (in m.p.g.): 35, 41, 36, 32, 40, 38, 38, 36. Do these results cast doubt on the advertised claim??
(a)Set up

and


(b) Determine the critical value(s) of

(c)Perform the statistical test.
(d)Draw final conclusions.

Question 11

The schizophrenia scale on a widely used personality scale is standardized to have a mean of 50 for a national group of normal adults. You administer the scale to a random sample of 20 students from a large university and obtain the following results:
Is there any reason to believe that the norms based on the national standardization group are inappropriate for college students?
(a)Set up

and

(b)Determine the critical value(s) of

(c)Perform the statistical test.
(d) Draw final conclusions.

Question 12

Translate each of the following statements into symbolic form involving a p-value:
(a) "The results did not reach significance at the .05 level."
(b)"The sample mean fell significantly below 50 at the .01 level."
(c)"The results were significant at the .001 level."
(d)"The difference between the sample mean and the hypothesized value of 100 barely missed significance at the .10 level."

Question 13

The answer to which of the following questions is the "key" to solving a statistical inference problem

A) are the sample values close to the population values?
B) what sample values are likely to occur under random sampling?
C) is the sample sufficiently representative of the population to proceed?
D) is the sample large enough?

Question 14

Statistic:parameter as

A) mean:standard deviation
B) sample:population
C) sample:mean
D) population:sample

Question 15

Which of the following is a parameter?

A)
B)
C) r
D) S

Question 16

The statistical hypothesis to be tested is called the

A) experimental hypothesis
B) normal hypothesis
C) population hypothesis
D) null hypothesis

Question 17

Which of the following has the characteristic of a null hypothesis?

A)
B)
C)
D) all of the above

Question 18

In testing a null hypothesis, the sample mean is compared with

A) sample means that would occur when the hypothesis is false
B) sample means that would occur when the hypothesis is true
C) various population means that would occur
D) means of samples for all possible values of n

Question 19

The null hypothesis is a statement that is

A) believed to be true until proven false
B) considered likely to be true
C) believed to be false until proven true
D) set up for the purpose of evaluating its truth or falsity

Question 20

In testing a hypothesis about a mean, the mean of the sampling distribution is taken to be

A) the value stated in
B) the value stated in
C) either of the above
D) neither of the above

Question 21

The region of rejection typically appears where in the sampling distribution?

A) above the mean
B) below the mean
C) in the center
D) at the extremes

Question 22

The criterion by which a decision is made about the null hypothesis is called

A) the level of significance
B) alpha
C) either of the above
D) the alternative hypothesis

Question 23

According to the information contained in the sampling distribution, we reject the null hypothesis if the probability of obtaining such a sample mean is

A) known
B) estimated
C) low
D) high

Question 24

When the average scores for two groups are found to be "statistically significant," it means that the results

A) are important
B) are of practical use
C) had a low probability of occurring by chance
D) all of the above

Question 25

If the sample mean is of the kind that could readily occur when the null hypothesis is true, we will

A) reject
B) retain
C) accept
as true
D) change the level of significance

Question 26

If
the null hypothesis will be

A) rejected 5% of the time
B) rejected 5% of the time when it is true
C) retained 5% of the time
D) retained 5% of the time when it is false

Question 27

We wish to test the null hypothesis that the mean is 80. If
is known, the normal curve model is appropriate when n is

A) 10
B) 25
C) greater than 50
D) any of the above

Question 28

If
and

and the critical values of z are

when

and

when

what would we do if our calculated value of z were +2.30?

A) retain
at

but reject it at

B) retain
at

and retain it at

C) reject
at

but retain it at

D) reject
at

and reject it at

Question 29

If
and

and the critical values of z are

when

and

when

what would we do if our calculated value of z were +1.80?

A) retain
at

but reject it at

B) retain
at

and retain it at

C) reject
at

but retain it at

D) reject
at

and reject it at

Question 30

If
and

and the critical values of z are

when

and

when

what would we do if our calculated value of z were -2.80?

A) retain
at

but reject it at

B) retain
at

and retain it at

C) reject
at

but retain it at

D) reject
at

and reject it at

Question 31

In ordinary small sample tests of a hypothesis about a single mean, the normal curve is not an adequate model for evaluating the outcome. The factor that is responsible is that

A)
varies too much from sample to sample
B) the sampling distribution of
is not normal for small samples
C)
is a variable
D)
is biased for small samples

Question 32

A biased estimator is one that yields an estimate that is

A) smaller than the quantity estimated
B) larger than the quantity estimated
C) either larger or smaller than the quantity estimated
D) in any instance larger or smaller than the quantity estimated, but that is on the average different from the quantity estimated

Question 33

S²

A) is always smaller than
B) tends, on the average, to be smaller than
C) is always larger than
D) tends, on the average, to be larger than

Question 34

The problem with using S as an estimate of
is that S

A) tends to be larger than
B) tends to be smaller than
C) decreases as n increases
D) increases as
increases

Question 35

When the population deviation is unknown, the standard error of the mean may be estimated by

A)
B)
C)
D) any of the above

Question 36

For inference about means, we need to know the value of the standard deviation of the population of scores. We estimate its value by calculating

A)
B)
C) either of the above
D) neither of the above

Question 37

For the following sample of three scores: 2, 4, 6, the best estimate of the population standard deviation is

A) 2
B)
C) 4
D)

Question 38

For the following sample of two scores: 1, 3, the best estimate of the population standard deviation is

A) 10
B) 1
C) 1.414
D) 1.732

Question 39

For the following sample of three scores: 1, 3, 5, the best estimate of the standard error of the mean is

A) 1.15
B) 1.73
C) 2.27
D) 2.52

Question 40

For a sample of 16 cases, the best estimate of the population standard deviation is calculated to be 8.36. The best estimate of the standard error of the mean (for samples of size 16) is

A) 2.16
B) .522
C) 2.09
D) 4.18

Question 41

If we were able to calculate both
and

we should

A) prefer
B) prefer
C) calculate either; they will be equivalent
D) prefer
for large samples and

for small samples

Question 42

Which of the following sets of values allows us to calculate t required to test a null hypothesis about a mean?

A)
B)
C)
D)

Question 43

Which statement, if any, is false?
The t distribution and the normal distribution of z are alike in that both

A) are symmetrical
B) have the same standard deviation
C) have the same mean
D) all of the above are true

Question 44

We conduct a two-tailed test at the .05 significance level with data that afford 8 degrees of freedom. When we look up the critical value of t, we will expect it to be ____________ the corresponding critical value of z.

A) substantially larger than
B) the same as
C) slightly smaller than
D) either larger than or smaller than, depending on n

Question 45

The mean of the t distribution

A) is
B) is the same as the mean of normally distributed z
C) varies with the size of the sample
D) varies with the distribution of scores from which it is derived

Question 46

The t distribution model is designed to correct for errors introduced when

A) the population standard deviation is estimated from the sample
B) sampling is not random
C) sampling is without replacement
D) the distribution of sample means is skewed

Question 47

Which statement, if any, is false? The distribution of t

A) has more area in its tails than does the normal distribution of z
B) varies in shape
C) depends on the number of degrees of freedom
D) all of the above are true

Question 48

In general, "degrees of freedom" is most closely related to the

A) level of significance
B) value of
C) value of
D) sample size

Question 49

When the number of degrees of freedom is very large,

A) z will be essentially equal to t
B) s will be very close to
C)
will be very close to

D) all of the above will be true

Question 50

As the number of degrees of freedom increases, student's distribution of t

A) remains the same
B) changes slowly at first, then more rapidly
C) changes rapidly at first, then more slowly
D) changes at an even rate

Question 51

For inference about single means, df is

A)
B)
C)
D)

Question 52

The region of rejection is determined by

A) the level of significance
B) the nature of the alternative hypothesis
C) both of the above
D) neither of the above

Question 53

Student's distribution of t has nothing to offer in obtaining relief from which problem, if any?

A) small sample size
B) unknown population standard deviations
C) nonnormality of the population distributions
D) it offers relief in all of the above situations

Question 54

We expect the alternative hypothesis to be a

A) particular value of a statistic
B) range of values of a statistic
C) particular value of a parameter
D) range of values of a parameter

Question 55

Which of the first three statements, if any, is false?
is

A) the hypothesis about which the statistical decision is made
B) always a statement about a parameter
C) always expressed as a point value
D) all of the above are true

Question 56

Which of the first three statements, if any, is false?
is

A) always a statement about a parameter
B) expressed as a range of values
C) states whether the test is directional or non-directional
D) all of the above are true

Question 57

The choice between a one-tailed test and a two-tailed test

A) will affect the way $H_0$ is stated
B) should be made after learning the location of X
C) is determined by the logic of the study rather than by the outcome of the data
D) is an optional matter; some statisticians prefer (b) and some prefer (c)

Question 58

If we test
against

the region of rejection will be located

A) at the extreme right
B) at the extreme left
C) at the two extremes
D) in the center

Question 59

We are thinking of a two-tailed test, but decide on a one-tailed test. This means that

A) the critical value of t will be numerically less
B) the critical value of t will be numerically greater
C) the value of $\alpha$ should be changed
D) the null hypothesis will be stated differently

Question 60

A one-tailed test should be used when

A) there is reason to believe the outcome will point in a particular direction
B) the experimenter so chooses; it is a matter of personal preference
C) the outcome of a test is in a particular direction
D) there is no practical difference in meaning between finding that the null hypothesis is true and finding that a difference exists in a direction opposite to that stated in the directional alternative hypothesis

Question 61

When a one-tailed test is employed (rather than a two-tailed test),

A) t must be numerically larger to result in rejection of the null hypothesis
B) a less discrepant value of $\overline{X}$ may result in rejection of the null hypothesis
C) the level of significance should be changed
D) n should be inversed

Question 62

Which, if any, need not be decided in advance of conducting the test of a null hypothesis?

A) the level of significance
B) the nature of
C) the region of rejection
D) all should be decided in advance

Question 63

When the value of
is changed from .05 to .01,

A) the standard error of the mean is reduced
B) the significance level is lowered
C) the more deviant a sample mean must be before it leads to rejection of the null hypothesis
D) both (b) and (c) are true

Question 64

When the value of
is changed from .05 to .01,

A) the greater the desirability of using a two-tailed test
B) the greater the value of t required to reject the null hypothesis
C) the smaller the region of retention
D) we should be less confident about our decision if that decision is to reject the null hypothesis

Question 65

When
is assigned a very small value,

A) we can be more certain of our conclusions, whatever the circumstances
B) there is greater risk of rejecting the null hypothesis when it is true
C) we should use a two-tailed test
D) we may overlook a real difference more easily than otherwise

Question 66

If the outcome of a test is significant at the .01 level, it

A) will also be significant at the .05 level
B) will not be significant at the .05 level
C) may be significant at the .05 level
D) will probably not be significant at the .05 level

Question 67

When
is assigned a relatively large value,

A) we can be more certain of our conclusions, whatever the circumstances
B) we should use a smaller n
C) we should use a two-tailed test
D) there is a greater risk of rejecting the null hypothesis when it is true

Question 68

An investigator retains all null hypotheses; some are true and some are false. What is
?

A) 1
B) 1/2
C) 0
D) cannot be determined

Question 69

We should probably use a lower value of
than usual when

A) n is small
B) we want to be quite confident that the null hypothesis should be rejected when we do so
C) we want to retain for further evaluation a possible (if not proven) difference
D) the test is directional

Question 70

If the outcome of a test is significant at the .05 level, it

A) will be significant at the .01 level
B) will not be significant at the .01 level
C) may be significant at the .01 level
D) will probably not be significant at the .01 level

Question 71

We should probably use a higher value of
than usual when

A) n is small
B) we want to be quite confident that the null hypothesis should be rejected when we do so
C) we want to retain for further consideration a possible (if not proven) difference
D) the test is directional

Question 72

When we use the .05 level of significance,

A) we will be right 5% of the time
B) we will be wrong 5% of the time
C) we will reject true null hypotheses 5% of the time
D) we will retain false null hypotheses 5% of the time

Question 73

When the criterion is the 5% significance level and the decision is to retain the null hypothesis, this implies that

A) the null hypothesis is true
B) the null hypothesis is probable true
C) the probability is .95 that the null hypothesis is true
D) the null hypothesis could be true, but so could a number of other null hypotheses

Question 74

Given:
If

is rejected, this means that

A) the null hypothesis is false
B) it seems likely that the null hypothesis is false
C) n was not large enough
D) the test should be repeated