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.
a. does this sample provide support for the hypothesis that the proportion of students who make partial payments that exceed the minimum amount has decreased since 2015?
determine the null and alternative hypotheses. choose the correct answer below.
○ a. $h_0: p > 0.23$
$h_1: p \leq 0.23$
○ b. $h_0: p \leq 0.23$
$h_1: p > 0.23$
○ c. $h_0: p = 0.23$
$h_1: p < 0.23$
○ d. $h_0: p = 0.23$
$h_1: p \
eq 0.23$
determine the critical value of the test statistic.
$z_\alpha = \square$
(type an integer or decimal rounded to two decimal places as needed.)
calculate the test statistic.
$z_0 = \square$
(type an integer or decimal rounded to two decimal places as needed.)

Explanation:

Step1: Determine the null and alternative hypotheses

The null hypothesis \(H_0\) is the statement of no change. The proportion in 2015 was \(p = 0.23\). The alternative hypothesis \(H_1\) is that the proportion has decreased. So \(H_0:p = 0.23\) and \(H_1:p<0.23\) (Option C).

Step2: Find the critical value

For a one - tailed test with \(\alpha=0.01\) and the alternative hypothesis \(p < 0.23\) (left - tailed), we look up the \(z\) - value in the standard normal distribution table. The critical value \(z_{\alpha}\) is the \(z\) - score such that \(P(Z<z_{\alpha})=\alpha\). From the standard normal table, \(z_{0.01}=- 2.33\).

Step3: Calculate the sample proportion

The sample proportion \(\hat{p}=\frac{x}{n}\), where \(x = 70\) (number of successes) and \(n = 380\) (sample size). So \(\hat{p}=\frac{70}{380}\approx0.1842\).

Step4: Calculate the test statistic

The formula for the test statistic \(z_0\) in a proportion test is \(z_0=\frac{\hat{p}-p}{\sqrt{\frac{p(1 - p)}{n}}}\). Substitute \(p = 0.23\), \(\hat{p}=0.1842\), and \(n = 380\) into the formula:

$$
LATEXBLOCK0
$$

Answer:

• Null and alternative hypotheses: \(H_0:p = 0.23\), \(H_1:p<0.23\) (Option C)
• Critical value: \(z_{\alpha}=-2.33\)
• Test statistic: \(z_0\approx - 2.12\)