Credit Card Ownership. Credit card ownership varies across age groups. In 2018, CreditCards.com estimated that the percentage of people who own at least one credit card is 67% in the 18–24 age group, 83% in the 25–34 age group, 76% in the 35–49 age group, and 78% in the 50+ age group. Suppose these estimates are based on 455 randomly selected people from each age group.
- a. Construct a 95% confidence interval for the proportion of people in each of these age groups who owns at least one credit card.
- b. Assuming the same
sample size will be used in each age group, how large would the sample need to be to ensure that the margin of error is .03 or less for each of the four confidence intervals?
a.
Find the 95% confidence interval for the population proportion of people in each of the given age groups who own at least one credit card.
Answer to Problem 63SE
The 95% confidence interval for the proportion of people in 18-24 age group who owns at least one credit card is
The 95% confidence interval for the proportion of people in 25-34 age group who owns at least one credit card is
The 95% confidence interval for the proportion of people in 35-49 age group who owns at least one credit card is
The 95% confidence interval for the proportion of people in 50+ age group who owns at least one credit card is
Explanation of Solution
Calculation:
The study is based on people from different age groups who own at least one credit card. There are 4 different age groups and in each group, 455 people are randomly selected.
For 18-24 age group, the percentage of people who owns at least one credit card is 67%.
For 25-34 age group, the percentage of people who owns at least one credit card is 83%.
For 35-49 age group, the percentage of people who owns at least one credit card is 76%.
For 50+ age group, the percentage of people who owns at least one credit card is 78%.
Margin of error for18-24 age group:
The formula for margin of error is as follows:
From “Table 8.1 value of
The value of margin of error is obtained as follows:
Confidence interval:
The 95% confidence interval for the proportion of people in 18-24 age group who owns at least one credit card is obtained as follows:
Thus, the 95% confidence interval for the proportion of people in 18-24 age group who owns at least one credit card is
Margin of error for25-34 age group:
From “Table 8.1 value of
The value of margin of error is obtained as follows:
Confidence interval:
The 95% confidence interval for the proportion of people in 25-34 age group who owns at least one credit card is obtained as follows:
Thus, the 95% confidence interval for the proportion of people in 25-34 age group who owns at least one credit card is
Margin of error for 35-49 age group:
From “Table 8.1 value of
The value of margin of error is obtained as follows:
Confidence interval:
The 95% confidence interval for the proportion of people in 35-49 age group who owns at least one credit card is obtained as follows:
Thus, the 95% confidence interval for the proportion of people in 35-49 age group who owns at least one credit card is
Margin of error for 50+ age group:
From “Table 8.1 value of
The value of margin of error is obtained as follows:
Confidence interval:
The 95% confidence interval for the proportion of people in 50+ age group who owns at least one credit card is obtained as follows:
Thus, the 95% confidence interval for the proportion of people in 50+ age group who owns at least one credit card is
b.
Find the number of sample needed for each of the four confidence intervals.
Answer to Problem 63SE
The number of sample needed for each of the four confidence intervals is 489.
Explanation of Solution
Calculation:
The margin of error is 0.03 and the value of
The formula for sample size for an interval estimate of a population proportion is as follows:
From the “Table 8.1 value of
The required sample size is as follows:
Thus, the sample size is 944.
Hence, the number of sample needed for each of the four confidence intervals 489.
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Chapter 8 Solutions
Essentials Of Statistics For Business & Economics
- Glencoe Algebra 1, Student Edition, 9780079039897...AlgebraISBN:9780079039897Author:CarterPublisher:McGraw HillCollege Algebra (MindTap Course List)AlgebraISBN:9781305652231Author:R. David Gustafson, Jeff HughesPublisher:Cengage Learning