Question

the majoring in money study by sallie mae from december 2015 showed that most college students manage their credit cards in a reasonable way. it showed 23% of students who own credit cards make partial payments that exceed the minimum amount due. you are curious to find out if the proportion of students who make partial payments that exceed the minimum amount has decreased since 2015 and collect data from a random sample of 380 students. you find that 70 students are making partial payments on their credit cards that exceed the minimum amount due. using \\( \alpha = 0.01 \\), complete parts a and b below. \\( z _ { 0 } = - 2.12 \\) (type an integer or decimal rounded to two decimal places as needed.) determine the conclusion. choose the correct answer below. a. reject \\( h _ { 0 } \\). there is sufficient evidence to conclude that the proportion of students who make partial payments that exceed the minimum amount has decreased since 2015. b. do not reject \\( h _ { 0 } \\). there is not sufficient evidence to conclude that the proportion of students who make partial payments that exceed the minimum amount has decreased since 2015. c. do not reject \\( h _ { 0 } \\). there is sufficient evidence to conclude that the proportion of students who make partial payments that exceed the minimum amount has decreased since 2015. d. reject \\( h _ { 0 } \\). there is not sufficient evidence to conclude that the proportion of students who make partial payments that exceed the minimum amount has decreased since 2015. b. determine the p - value for this test.

Explanation:

Step1: Find the critical value

For a one - tailed test with \(\alpha = 0.01\), the critical value \(z_{\alpha}\) is \(z_{0.01}=- 2.33\) (since it is a left - tailed test, we look at the negative \(z\) - value). The test statistic \(z_0=-2.12\).

Step2: Compare the test statistic and the critical value

We know that if \(z_0-2.33\) (i.e., \(z_0\) is not in the rejection region).

Step1: Use the standard normal distribution table

The \(p\) - value for a left - tailed test with \(z=-2.12\) is found by looking up the value of \(\Phi(-2.12)\) in the standard normal distribution table.

Step2: Look up the value

From the standard normal table, \(\Phi(-2.12) = 0.0170\)

Answer:

B. Do not reject \(H_0\). There is not sufficient evidence to conclude that the proportion of students who make partial payments that exceed the minimum amount has decreased since 2015.

For part b: