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Problem 16 Easy Difficulty

In a study entitled How Undergraduate Students Use Credit Cards, it was reported that undergraduate students have a mean credit card balance of $\$ 3173$ (Sallie Mae, April 2009 ). This figure was an all-time high and had increased 44$\%$ over the previous five years. Assume that a current study is being conducted to determine if it can be concluded that the mean credit card balance for undergrate students has continued to increase compared to the April 2009 report. Based on previous studies, use a population standard deviation $\sigma=\$ 1000.$
a. State the null and altemative hypotheses.
b. What is the $p$ -value for a sample of 180 undergraduate students with a sample mean credit card balance of $\$ 3325 ?$
c. Using a .05 level of significance, what is your conclusion?

Answer

a. $H_{0} : \mu=3173, H_{a} : \mu>3173$
b. $P=0.0207$
c. There is sufficient evidence to support the claim of an increase.

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Video Transcript

This is a basic question. Now move your party. What is going to be Arnel hypothesis? Um, null hypothesis in this case will become that my immune is less than or equal to 3173 Right. And my alternative hypothesis will become what it will become that my new is greater than 3173 Okay, now moving on the part B. Right? The party says that I want to find the p value for a sample of 1 80 undergraduate students with the sample mean off 33 to 5. All right, so I'm going to use the Z statistic in this case. So what? How do we find the Z value? It is our sample mean, which happens to be 33 to 5, minus the hypothesized mean, which happens to be 3173 Upon the standard deviation that we have, that is 1000, right? The population standard deviation upon route off the sample size, which happens to be 1 80. Okay, Now this part of the denominator happens to be the best estimator for our sample standard deviation. So hence we're doing this. And this is also the formula. You can check in the textbook. Now, if I use my calculated to find this, this value turns out to be 2.39 My Z sadistic Turns out to be 2.39 Now, how do you find the P value? I have my Z statistic. I can use any statistical software or any online tool to find my p value. Just input the zed value and you get the P value. And in this case, I am getting my P value as zero point 0207010207 Okay, what is part C. Parsi says we have to use this 0.5 level of significance and conclude if I use 0.5 level of significance, it means that my Alfa is 0.0 fight and I can see that my P is less than Alfa Hence, my result is statistically significant, so I can reject my It's not I can reject. It's not okay, So if we go back, what does that It's not It's not once that mu is less than equal toe. Three point. Sorry. 3173 So, yes, I can say that I have enough statistical evidence to suggest that there has been an increase in the mean balance and it is more than 3173 So, yes, we can say that we have enough evidence. Sufficient statistical evidence to support the claim right to support the claim often increase. This is how we go about doing this question.