AP Statistics 3.7: Proportion Hypothesis Tests

Section
1

Let p represent the proportion of all customers of the pet supply company who would place an order within 30 days after receiving an e-mail with a coupon for $10 off the next purchase.

The null hypothesis is $\mathrm{H}_{0}: p=0.40$, and the alternative hypothesis is $\mathrm{H}_{\mathrm{a}}: p>0.40$.

An appropriate test is a one-sample z-test for a population proportion.

Essentially correct (E) if the response satisfies the following three components:
1. States the correct equality for the null
hypothesis for a proportion (e.g., p=0.40 )
AND the correct direction of the one-sided alternative hypothesis for a proportion (e.g.,
p>0.40 )
2. Provides sufficient context for the parameter
by including reference to the population proportion AND the sampling units
(customers) AND the response variable
(placing an order after receiving a coupon)
3. Identifies a one-sample z-test for a
population proportion by name (e.g., "one-proportion z-test" but not merely "one-sample z-test") or by formula
Partially correct (P) if the response does not
meet the criteria for E but satisfies either
component 1 and/or component 3.
Incorrect (I) if the response does not meet the
criteria for E or P.

- The elements of component 2 do not have to be satisfied with the statement of the hypotheses. They may
be satisfied by work presented anywhere in the response, most likely by the statement of the conclusion.
- If the statement of the hypotheses refers to population proportion and the conclusion refers to sample proportion (or vice versa), then the population aspect of component 2 is not satisfied.

- A response that states the null hypothesis as $\mathrm{H}_{0}: p \leq 0.40$ may satisfy component 1.

- To satisfy component 1, the hypotheses must be stated in terms of a proportion. If a symbol other than p or $\pi$ is used to denote the proportion, it must be clearly defined as a proportion (but does not need to reflect the context of customers who would place an order within 30 days after receiving a coupon) in order for the response to satisfy component 1. It is acceptable to use " $p_{0}$ " to denote the proportion.

- A response that states the hypotheses in words (e.g., "the null hypothesis is that the proportion is 0.40, and the alternative hypothesis is that the proportion is greater than $0.40^{\prime \prime}$ ) may satisfy component 1. Neither context nor the concept of the population is required to satisfy component 1.

- A response that states the hypotheses in words (e.g., "the null hypothesis is that the proportion of all customers who would place an order within 30 days after receiving a coupon is equal to 0.40, and the alternative hypothesis is that the proportion is greater than $0.40^{\prime \prime}$ ) may satisfy component 1 and component 2.

- If the response clearly refers to the sample proportion instead of the population proportion using words or a symbol (e.g., $\hat{p}$ ), then component 2 is not satisfied unless the symbol used is defined as the population proportion.

- A response may satisfy the population aspect of component 2 by doing the following:

referring to population in the statement of the conclusion of the inferential procedure

using notation such as $p, p_{0}$, or $\pi$ when defining the hypothesis statements.

- A response may satisfy the sampling units aspect of component 2 by referring to "people who place an order" or similar statement.

- If the response identifies the correct test by name, but also states an incorrect formula, then component 3 is not satisfied.

- If the response identifies the test by formula using a t-percentile instead of a z-percentile, then component 3 is not satisfied.

Confidence Interval Approach:

- If a one-sample z-interval for a population proportion is identified correctly by name (e.g., "one

proportion z-interval" but not merely "one-sample z-interval") or by formula, then component 3 is satisfied.

- If a response uses a one-sample z-interval for a population proportion, then component 2 is satisfied if the response indicates that it is a confidence interval for the proportion of all customers who would place an order within 30 days after receiving a coupon, even if the hypotheses are not stated.

Model Solution

Scoring

(a)
Section
2

The independent observations condition for performing the one-sample z-test for a population proportion is satisfied because the data were obtained from a random sample of 90 customers who placed an order in the past year and, because sampling of customers is done without replacement, it is assumed that this large online company has more than 10(90)=900 customers.

The sample size is large enough to support an assumption that the sampling distribution of $\hat{p}$ is approximately normal because (90)(0.4)=36 and (90)(1-0.4)=54 are both at least 10.

The value of the sample proportion is $\hat{p}=\frac{38}{90} \approx 0.422$ and the value of the test statistic is $z=\frac{\frac{38}{90}-0.40}{\sqrt{\frac{(0.40)(0.60)}{90}}} \approx 0.430$. The corresponding p-value is
$$P(z>0.430) \approx 0.333.$$

Essentially correct (E) if the response satisfies the following four components:
1. Checks the independence condition by
referring to the random selection of
90 customers AND indicating that the
company is assumed to have at least
900 customers (i.e., $90 \leq 0.10 N$ )
2. Checks that the sample size is large enough
to support the assumption that the sampling
distribution of $\hat{p}$ is approximately normal
by verifying that (90)(0.4) and
(90)(1-0.4) are both at least 10 (or 5)
4. Correctly reports the p-value, consistent with the reported test statistic and stated alternative hypothesis
Partially correct (P) if the response satisfies only two or three of the four components.
Incorrect (I) if the response does not meet the
criteria for E or P.

Additional Notes:

- In order to satisfy the reference to the random selection of 90 customers in component 1 it is minimally acceptable to state "random sample - check" or "SRS - check." However, component 1 is not satisfied if the response implies that random assignment was used or only states "random - check."

- In order to satisfy component 2, the response must include actual values of the observed successes and failures, or values for the expected successes and failures, or formulas for the expected number of successes and failures with values inserted AND the response must make a comparison of the two values with some standard criterion, such as 5 or 10. If expressions such as (90)(0.4) and (90)(1-0.4) are used, simplification is not required.

Examples of acceptable quantities (comparisons must still be made):

- 38 and 52 (observed counts)

- 36 and 54 (expected counts under the null hypothesis)

- (90)(0.4) and (90)(1-0.4)

- (90)(0.4222) and (90)(1-0.422)

Unless values of all parameters are explicitly defined in the response, the following quantities are unacceptable:

- 90 p, 90(1-p), n p, n(1-p)

- $90 \hat{p}, 90(1-\hat{p}), n \hat{p}, n(1-\hat{p})$

- n(0.4), n(0.6), n(0.4222), n(1-0.4222)

- When computing the test statistic, using a $\hat{p}$ of 0.42 in the numerator results in a test statistic equal to 0.39 with a p-value of 0.36. These values satisfy components 3 and 4.

- If the response uses $\hat{p}$ in the null standard error formula to calculate the z-statistic, component 3 may be satisfied.

- A response that reports the correct value for the z-statistic but contains errors in supporting work may still satisfy component 3.

- If the response satisfies component 4, any supporting work for the p-value may be treated as extraneous.

- If the response compares the value of the test statistic to a critical value instead of reporting a p-value, then the critical value (1.645), or a critical value consistent with the stated alternative hypothesis, satisfies component 4.

- If a two-tailed alternative hypothesis is stated or the direction of the stated one-tailed alternative hypothesis is incorrect, then the p-value must be consistent with the stated alternative hypothesis to satisfy component 4.

- If the response omits identifying the hypotheses, the correct one-sided alternative hypothesis is assumed when scoring component 4.

- If an incorrect alternative hypothesis is stated, then the p-value must be consistent with the stated alternative hypothesis to satisfy component 4.

Confidence Interval Approach:

- If either a one-sided 95 percent confidence interval is correctly calculated as (0.337, 1), or a two-sided 90 percent confidence interval is correctly calculated as ( 0.337,0.508 ), then component 3 is satisfied.

- If only the lower end of a confidence interval is used to reach a conclusion, then component 4 is satisfied.

- Application of a confidence interval approach must be consistent with the stated alternative to satisfy component 4. A two-sided 95 percent confidence interval is (0.320, 0.524), and a lower one-sided 95 percent confidence interval is (0, 0.508).

Model Solution

Scoring

(a)
Section
3

Because the p-value is greater than $\alpha=0.05$,
the null hypothesis should not be rejected. The results from this study do not provide
convincing statistical evidence that the
manager's belief is correct. That is, there is not
convincing statistical evidence that more than 40 percent of all customers of the pet supply
company would place an order within 30 days
after receiving an e-mail with a coupon for $10
off the next purchase.

Essentially correct (E) if the response satisfies the following two components:
1. Provides a correct justification of the
conclusion based on whether the p-value is
less than $\alpha=0.05$ (or a comparison of the
value of the test statistic to an appropriate
critical value, e.g., z<-1.645 )
2. States a correct conclusion consistent with the stated alternative hypothesis OR states a
conclusion that answers the inference question (e.g., states the conclusion in terms
of the manager's belief)
Partially correct (P) if the response satisfies only one of the two components.
Incorrect (I) if the response does not meet the
criteria for E or P.

Additional Notes:

- Although including proper context (the concept of population proportion and referencing the response variable) is important in stating the conclusion, context displayed in stating the conclusion is considered in scoring component 2 of Section 1.

- The response need not make an explicit decision about the null hypothesis (reject $\mathrm{H}_{0}$ or fail to reject $\mathrm{H}_{0}$ ) in order to satisfy component 1. However, if an incorrect decision is stated, then component 1 is not satisfied.

- If the conclusion and justification are consistent with an incorrect p-value (or an incorrect value of the test statistic, or an incorrect confidence interval), the response may satisfy component 1 and component 2.

- If the response omits hypotheses, assume the correct alternative hypothesis, $\mathrm{H}_{\mathrm{a}}: p>0.40$, was provided when scoring component 1 and component 2.

- If the conclusion includes a definitive statement (e.g., "this proves that we do not have enough evidence to claim..." or "there is no evidence..."), then component 2 is not satisfied.

- If the response includes a statement that is equivalent to accepting the null hypothesis (e.g., "we conclude that the proportion of customers who will place an order is 0.40 "), then component 2 is not satisfied.

- If the response includes an incorrect interpretation of the p-value, then the score for Section 3 is lowered one level (that is, from E to P or from P to I).

- The clarity and quality of the statement of the conclusion and the statement of the justification may be used in a holistic approach to decide whether to score up or down (e.g., raising a score of 2.5 to 3 or reducing a score of 2.5 to 2).

Confidence Interval Approach:

- If the alternative hypothesis is specified correctly as $\mathrm{H}_{\mathrm{a}}: p>0.40$, then component 1 is satisfied if the justification is based on whether 0.40 is below the lower end of the confidence interval. If the alternative hypothesis is stated in the wrong direction, then component 1 is satisfied if the justification is based on whether 0.40 is above the upper end of the confidence interval.

- If no alternative hypothesis is specified in the response, then assume the correct alternative hypothesis is provided when scoring component 2.

- If an incorrect two-sided alternative hypothesis is specified, then component 2 is satisfied if the justification is based on whether 0.40 is included in the confidence interval.

- If the response includes an incorrect interpretation of the confidence interval, then the score for Section 3 is lowered one level (that is, from E to P or from P to I).

Model Solution

Scoring

Model Solution and scoring sections

Section 1 — Procedure and hypotheses
Use a one-sample z-test for a population proportion. Let p be the true proportion of students at Karen's high school who use the app for homework at least once per week.
$$H_0:p=0.22,\qquad H_a:p>0.22.$$
An essentially correct section identifies the one-proportion z-test, states both hypotheses with the correct value and direction, and defines the parameter in context.

Section 2 — Conditions and calculations
The sample is a simple random sample of 130 students. Independence is reasonable because 130<0.10(2000). The large-counts condition holds because
$$130(0.22)=28.6\ge10,\qquad130(0.78)=101.4\ge10.$$
Also,
$$\hat p=\frac{38}{130}\approx0.2923,$$$$z=\frac{0.2923-0.22}{\sqrt{0.22(1-0.22)/130}}\approx1.99,\qquad P(Z>1.99)\approx0.023.$$
An essentially correct section checks independence and large counts and reports a correct test statistic and consistent one-sided p-value.

Section 3 — Conclusion
Because 0.023<0.05, reject $H_0$. There is convincing statistical evidence that the population proportion of students at Karen's school who use the app at least weekly for homework is greater than Country W's proportion of 0.22. An essentially correct section compares the p-value with $\alpha$, makes a consistent decision, and gives a nondefinitive conclusion in context.

Overall scoring: three essentially correct sections earn 4; two E and one P earn 3; the official holistic table determines scores 2 and 1 from the remaining E/P combinations.