Got Sleep?
Question #4
Intent of the Question
The primary goal of this question was to assess a
student's ability to identify, set up, perform, and
interpret the results of an appropriate hypothesis
test to address a particular question.
More specific goals were to assess a student's
ability to (1) state appropriate hypotheses; (2)
identify the appropriate statistical test procedure
and check appropriate conditions for inference; (3)
calculate the appropriate test statistic and p-value;
and (4) draw an appropriate conclusion, with
justification, in the context of the study.
Question #4
The National Sleep Foundation conducts an annual survey to track sleep
related behaviors of U.S. adults. In their most recent survey, a random sample
of 1,018 adults answered the question "About how much actual sleep would
you estimate you typically get on work nights or weeknights?" The frequency
table below summarizes the responses by whether they were less than, equal
to, or more than the recommended 7 to 9 hours of sleep and by the age
group of the respondent.
Typical Weeknight Sleep Time
Age Group
Less Than
7 Hours
Recommended
7 to 9 Hours
More Than
9 Hours
Total
18 – 34
44
78
109
231
35 – 54
144
200
218
562
55 or older
47
77
101
225
Total
235
355
428
1,018
At the 𝛼 = 0.05 significance level, do the data provide convincing statistical
evidence that there is an association between age group and typical
weeknight sleep time for adults in the United States?
Solution Step 1
States a correct pair of hypotheses.
The null hypothesis is that typical weeknight sleep time is
independent of (that is, it is not associated with) age
group for the population of adults in the United States.
The alternative hypothesis is that typical weeknight sleep
time is not independent of (that is, it is associated with)
age group for the population of adults in the United
States.
Scoring Step 1
Each of steps 1, 2, 3, and 4 are scored as essentially correct (E), partially
correct (P), or incorrect (I).
Step 1 is scored as follows:
Essentially correct (E) if the response correctly states both hypotheses with at
least one in context.
Partially correct (P) if the response correctly states both hypotheses but not
in context,
OR
the hypotheses were reversed with at least one stated in context.
Incorrect (I) if the response does not meet the criteria for E or P.
Note: If the hypotheses contain language that suggests that the response
refers to the sample data (for example, by referring to the adults that
answered the question), step 1 is scored as incorrect (I).
Solution Step 2
Identifies a correct test procedure (by name or by
formula) and checks appropriate conditions.
The appropriate test is a chi-square test of independence.
The conditions for this test were satisfied because:
The question states that the sample was randomly
selected.
The expected counts for all nine cells of the table
were at least 5, as seen in the following table that
lists expected counts in parentheses beside the
observed counts:
Age Group
18 – 34
35 – 54
55 or older
Total
Less Than
7 Hours
44 (53.3)
144 (129.7)
47 (51.9)
235
Recommended
7 to 9 Hours
78 (80.6)
200 (196.0)
77 (78.5)
355
More Than
9 Hours
109 (97.1)
218 (236.3)
101 (94.6)
428
Total
231
562
225
1,018
Scoring Step 2
Essentially correct (E) if the response correctly includes the following three components:
Identifies a chi-square test of independence by name or by formula for the chi-square test statistic.
States AND verifies the random sampling condition.
States AND verifies the technical condition that all expected counts are greater than 5.
Partially correct (P) if the response correctly includes two of the three components listed above.
Incorrect (I) if the response does not meet the criteria for E or P.
Notes:
• If the response identifies the test procedure as a chi-square test of homogeneity of proportions,
step 2 does not receive credit for component 1.
• If the response identifies the correct test procedure but gives an incorrect formula for the test
statistic, then this is considered a contradiction and does not meet the criteria for component 1.
• Stating the condition that the expected counts must be greater than 5 is not in itself sufficient for
satisfying component 3; the condition must be checked by reporting expected counts, or minimally
reporting the value of the smallest expected count and indicating that it is at least 5.
• If the response includes an incorrect technical condition such as "n ≥ 30" or "normality," then this
will be considered a parallel solution and credit will not be granted for component 3.
• If the response states and verifies the condition that 80 percent of all expected counts must be ≥ 5
and all expected counts must be ≥ 1, then the response can receive credit for component
Solution Step 3
Correct mechanics, including the value of the test statistic and
p-value (or rejection region).
2
The test statistic
is
calculated
from
𝜒
𝑂−𝐸 2
=
; that is,
𝐸
𝜒2
= 1.631 + 0.081 + 1.453 + 1.569 + 0.082 + 1.415 + 0.470
+ 0.027 + 0.433 = 7.161.
The p-value is P 𝜒 2 ≥ 7.161 = 0.1276
based on 3 − 1 × 3 − 1 = 4 degrees of freedom.
Scoring Step 3
Step 3 is scored as follows:
Essentially correct (E) if the response correctly calculates the following two
components:
Test statistic
p-value or critical value
Partially correct (P) if the response correctly calculates one of the two
components listed above.
Incorrect (I) if the response does not meet the criteria for E or P.
Notes:
When a response has an error in one calculation, future calculations are
considered correct if they follow from the initial miscalculation.
The correct critical value is 9.49 for a significance level of 0.05.
Solution Step 4
States a correct conclusion in the context of the
study, using the result of the statistical test.
Because the p-value is greater than the given
significance level of 𝛼 = 0.05, we fail to reject H0.
The data do not provide enough evidence at the
0.05 level of significance to conclude that there is
an association between age group and typical
weeknight sleep time for adults in the United
States.
Scoring Step 4
Essentially correct (E) if the response provides a correct
conclusion in context, with justification based on linkage
between the p-value and the given 𝛼 = 0.05.
Partially correct (P) if the response provides a correct
conclusion, with linkage to the p-value, but not in context;
OR
if the response provides a correct conclusion in context, but
without justification based on linkage to the p-value.
Incorrect (I) if the response does not meet the criteria
for E or P.
Notes Section 4:
The conclusion must be consistent with the hypotheses.
The conclusion must be related to the alternative hypothesis.
If the p-value is incorrect, step 4 is scored as E if the response includes proper linkage and a
conclusion in context consistent with that p-value.
If the p-value is incorrect and less than 0.05, wording that states or implies the alternative
hypothesis is proven lowers the score one level (that is, from E to P or P to I) in step 4.
If the p-value is greater than 0.05, wording that states or implies that the null hypothesis is
accepted lowers the score one level (that is, from E to P or P to I) in step 4.
A response including incorrect statistical language (for example, discussing the correlation instead
of the association between the two variables) lowers the score one level (that is, from E to P or P
to I) in step 4.
Since 𝛼 = 0.05 is given, explicit linkage between the size of the p-value and 𝛼 = 0.05 is required
(for example, stating that the p-value is large is not sufficient).
Scoring
Each essentially correct (E) step counts as 1 point. Each
partially correct (P) step counts as ½ point.
4
Complete Response
3
Substantial Response
2
Developing Response
1
Minimal Response
If a response is between two scores (for example, 2½ points),
score down.