Question

a credit card company claims that the mean credit card debt for individuals is greater than $4,900. you want to test this claim. you find that a random sample of 34 cardholders has a mean credit card balance of $5,090 and a standard deviation of $625. at \\( \alpha = 0.05 \\), can you support the claim? complete parts (a) through (e) below. assume the population is normally distributed. (a) write the claim mathematically and identify \\( h _ { 0 } \\) and \\( h _ { a } \\). which of the following correctly states \\( h _ { 0 } \\) and \\( h _ { a } \\)? \\( \square \\) a. \\( h _ { 0 } : \mu \
eq 4,900 \\) \\( h _ { a } : \mu \leq 4,900 \\) \\( \square \\) b. \\( h _ { 0 } : \mu > 4,900 \\) \\( h _ { a } : \mu \leq 4,000 \\) \\( \square \\) c. \\( h _ { 0 } : \mu \leq 4,900 \\) \\( h _ { a } : \mu > 4,900 \\) \\( \square \\) d. \\( h _ { 0 } : \mu \geq 4,900 \\) \\( h _ { a } : \mu < 4,900 \\) \\( \square \\) e. \\( h _ { 0 } : \mu = 4,900 \\) \\( h _ { a } : \mu > 4,900 \\) \\( \square \\) f. \\( h _ { 0 } : \mu = 4,900 \\) \\( h _ { a } : \mu \
eq 4,900 \\) (b) find the critical value(s) and identify the rejection region(s) what is(are) the critical value(s), \\( t _ { 0 } \\)? (use a comma to separate answers as needed. round to three decimal places as needed.)

Explanation:

Step1: Understand the null and alternative hypotheses

The null hypothesis \(H_0\) is a statement of equality or non - effect. The alternative hypothesis \(H_a\) is the claim we are trying to find evidence for. The claim is that the mean credit card debt \(\mu\) is greater than \(\$4900\). So, \(H_0:\mu\leq4900\) (the opposite of the claim for the null) and \(H_a:\mu > 4900\) (the claim).

Step2: Determine the critical value

Since the population standard deviation \(\sigma\) is unknown (we are given the sample standard deviation \(s = 825\)), and the sample size \(n=34\) (\(n<30\) would be a small sample, but here \(n = 34\) and population is normally distributed), we use the \(t\) - distribution. The significance level \(\alpha=0.05\) and the degrees of freedom \(df=n - 1=34-1 = 33\).
Using a \(t\) - table or a calculator, the critical value \(t_0\) for a right - tailed test with \(\alpha = 0.05\) and \(df=33\) is \(t_0=1.692\). The rejection region is \(t>1.692\).

Answer:

(a) The correct pair is \(H_0:\mu\leq4900\) and \(H_a:\mu > 4900\) (Option B).
(b) The critical value \(t_0 = 1.692\) and the rejection region is \(t>1.692\).