3.3 Constructing a Confidence Interval for a Population Proportion question 1
[Maximum number: 4]
A polling agency showed the following two statements to a random sample of 1,048 adults in the United States.
Environment statement: Protection of the environment should be given priority over economic growth. Economy statement: Economic growth should be given priority over protection of the environment.
The order in which the statements were shown was randomly selected for each person in the sample. After reading the statements, each person was asked to choose the statement that was most consistent with his or her opinion. The results are shown in the table.
Question (a)
(a)
Assume the conditions for inference have been met. Construct and interpret a 95 percent confidence interval for the proportion of all adults in the United States who would have chosen the economy statement.
Part (a):
The appropriate procedure is a one-sample z-interval for a population proportion. The problem stated the conditions for inference have been met, so they do not need to be checked. A 95 percent confidence interval for the population proportion is given as p^±z∗np^(1−p^), which is
We are 95 percent confident that the population proportion of all adults in the U.S. who would have chosen the economy statement is between 0.34 and 0.40.
Question (b)
(b)
One of the conditions for inference that was met is that the number who chose the economy statement and the number who did not choose the economy statement are both greater than 10. Explain why it is necessary to satisfy that condition.
Part (b):
The condition is necessary because the formula for the confidence interval relies on the fact that the binomial distribution can be approximated by a normal distribution which then results in the sampling distribution of p^ being approximately normal. The approximation does not work well unless both np^ and n(1−p^) are at least 10.
Question (c)
(c)
A suggestion was made to use a two-sample z-interval for a difference between proportions to investigate whether the difference in proportions between adults in the United States who would have chosen the environment statement and adults in the United States who would have chosen the economy statement is statistically significant. Is the two-sample z-interval for a difference between proportions an appropriate procedure to investigate the difference? Justify your answer.
Part (c):
The suggested procedure is not appropriate because one of the requirements for using a two-sample z-interval for a difference between proportions is that the two proportions are based on two independent samples. In the situation described the two proportions come from a single sample and thus are not independent.
Scoring
This question is scored in three sections. Section 1 consists of the mechanics of the confidence interval in part (a), section 2 consists of interpreting the confidence interval in part (a), and section 3 consists of parts (b) and (c).
Section 1 is scored as follows: Essentially correct (E) if the response includes the following two components:
States the correct procedure by name or formula
Calculates the confidence interval
Partially correct (P) if the response includes only one of the two components. Incorrect (I) if the response does not meet the criteria for E or P.
Notes:
- A formula with correct values is sufficient for component 2.
- Component 2 can never be satisfied if the response contains an incorrect formula.
Section 2 is scored as follows: Essentially correct (E) if the response includes the following four components for the interval interpretation:
Estimates a proportion
Infers about the population
States 95 percent confidence
Includes context
Partially correct (P) if the response satisfies components 1 and 2 AND satisfies only one of components 3 or 4;
OR if the response gives a correct interpretation of the confidence level in context without interpreting the specific interval.
Incorrect (I) if the response does not meet the criteria for E or P. Notes:
- Any indication of an inference to the adults sampled rather than the population does not satisfy component 2 and is scored as I.
- Stating values that are unrealistic as proportions (including blanks) in the interpretation lowers the score by one level (from E to P or from P to I).
- When both the interpretation of the interval and the level are given, only the interpretation of the interval is scored.
Section 3 is scored as follows: Essentially correct (E) if the response includes the following two components:
Part (b) states that the condition implies the sampling distribution of p^ is approximately normal OR the normal approximation to the binomial distribution is appropriate.
Part (c) indicates the procedure is not appropriate because the two proportions come from a single sample (dependent) rather than two (independent) samples.
Partially correct (P) if the response includes only one of the two components. Incorrect (I) if the response does not meet the criteria for E or P. Notes:
- Referring to the sampling distribution needing to be normal without explicitly stating p^ satisfies component 1.
- Referring to normal approximation must specify "to binomial" to satisfy component 1.
- Discussing the two-sample z-interval versus the two-proportion z-interval is extraneous information.
- In part (c) component 2 can never be satisfied if a procedure that is not possible for one sample of categorical data is suggested.
Complete Response Three parts essentially correct
Substantial Response Two parts essentially correct and one part partially correct
Developing Response Two parts essentially correct and no parts partially correct
OR One part essentially correct and one or two parts partially correct
OR Three parts partially correct
Minimal Response One part essentially correct
OR No parts essentially correct and two parts partially correct
Intent of Question
The primary goals of this question were to assess a student's ability to (1) use a scatterplot to comment on a report about the relationship between two variables and interpret the slope for the least-squares regression line summarizing this relationship; (2) describe the relationship between two variables in a scatterplot when a categorical variable is introduced and compare a characteristic of the distribution of a variable for different categories of individuals in a scatterplot; and (3) describe how the associations between two variables for each category of individuals in a scatterplot differ from the overall association in the same scatterplot.
Solution
3.3 Constructing a Confidence Interval for a Population Proportion question 2
A survey conducted by a national research center asked a random sample of 920 teenagers in the United States how often they use a video streaming service. From the sample, 59% answered that they use a video streaming service every day.
Construct and interpret a 95\% confidence interval for the proportion of all teenagers in the United States who would respond that they use a video streaming service every day.
Section 1
The appropriate procedure is a one-sample z-interval for the proportion of all teenagers in the United States who would respond that they use a video streaming service every day.
Essentially correct (E) if the response satisfies the following two components: 1. Identifies the appropriate procedure as a one-sample z-interval by name or formula or by the calculations of the correct confidence interval endpoint values 2. States that the parameter of interest is the population proportion 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:
- The response to component 2 concerning the statement of "population proportion" can be found in any of the three sections of part (a).
- Any notation used to represent sample proportion or population proportion should remain consistent throughout part (a).
Model Solution
Scoring
(a) Section 2
This survey selected a random sample of 920 teenagers in the United States, which enables the interval to be generalized to the population of interest. This sample of 920 teenagers is less than 10\% of the total number of teenagers in the United States, which is required as sampling was conducted without replacement from a finite population. In addition, there were more than 10 successes and 10 failures as (920)(0.59)=542.8( or 543) responded that they use a streaming service daily and (920)(0.41)=377.2 (or 377) responded that they did not. Thus, the sample size is large enough to support the assumption that the sampling distribution of p^ is approximately normal.
Therefore, a 95\% confidence interval for the population proportion is given by p^±z∗np^(1−p^)=0.59±1.96920(0.59)(0.41), which is 0.59±0.032, and the interval is (0.558, 0.622).
Essentially correct (E) if the response satisfies the following four components: 2. Indicates 920 is less than ten percent of all teenagers in the United States 3. Verifies that there are at least 10 successes and failures by calculating the following values np^=(920)(0.59)≈542.8 and n(1−p^)=(920)(1−0.59)≈377.2 4. Reports the values for a correct interval consistent with the procedure stated in Section 1 Partially correct (P) if the response satisfies either component 3 or component 4 and at least one of the other three components. Incorrect (I) if the response does not meet the criteria for E or P.
Additional Notes:
- Stating the large sample condition without verification is not sufficient for component 3.
- If the response includes an inappropriate check of conditions, such as n>30, then component 3 is not satisfied.
- Supporting work, showing formulas or calculations, is not required for component 4.
- If the interval values are correct, the use of a one-sample z procedure for proportion can be used to satisfy component 1 of Section 1.
- A response that uses the value of x=543 will result in an interval of (0.5584, 0.6219), and a response that uses the value of x=542 will result in an interval of (0.5573, 0.6209). These interval endpoint values may be used to satisfy component 4.
- If the response includes supporting work for calculating the confidence interval that displays a correct formula with correct values inserted for p^,n, and z, then component 4 is satisfied even if values for the endpoints of the confidence interval are not displayed or calculated incorrectly.
- A response that computes an interval in percentages rather than proportions may satisfy component 4 if the response correctly indicates the use of percentages, (55.8\%, 62.2\%).
- Minor errors or omissions when checking assumptions may be considered if holistic scoring is required.
Model Solution
Scoring
(a) Section 3
We can be 95\% confident that the proportion of all teenagers in the United States who would respond that they use a streaming service every day is between 0.558 and 0.622.
Essentially correct (E) if the response satisfies the following two components: 1. Indicates 95\% confidence and interprets the interval using words such as "we are 95\% confident" or "with 95\% confidence" and provides interval endpoint values consistent with calculations in Section 2 2. Conveys inference about a population proportion in the proper context, i.e., "the true proportion of U.S. teenagers," "the population proportion of U.S. teenagers," or "the proportion of all U.S. teenagers" 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:
- For component 1, responses that use the same incorrect proportion interval endpoint values calculated in Section 2 satisfies this portion of component 1.
- Section 3 is scored I if the response includes unreasonable values for a proportion or percentage.
- Responses must be consistent with the use of the terms proportion and percentage in component 2. A response that refers to the population percentage must use percentage values to satisfy component 2, and a response that refers to the population proportion must use proportion values.
- Any interpretation of the level of the interval, whether correct or incorrect, is considered extraneous and does not satisfy component 1.
- Clear indication of an inference to the sample of 920 teenagers, rather than the population of teenagers, does not satisfy component 2.
- Section 3 is scored I if it implies that the population proportion is a random variable. For example, it indicates that there is a 95 percent chance that the population proportion is between 0.558 and 0.622.
- A response that implies that we have found the actual percentage of U.S. teenagers who respond that they use a streaming service (deterministic) does not satisfy component 1.
