The student academic group on a college campus claims that...

The student academic group on a college campus claims that freshman students study at least 2.5 hours per day, on average. One Introduction to Statistics class was skeptical. The class took a random sample of 30 freshman students and found a mean study time of 137 minutes with a standard deviation of 45 minutes. At $\alpha=0.01$ level, is the student academic group's claim correct?

Key Concepts

p-Value

The p-value is the probability, under the assumption that the null hypothesis is true, of obtaining a test statistic at least as extreme as the one observed. It provides a measure of the strength of the evidence against the null hypothesis; if the p-value is less than the significance level, the null hypothesis is rejected in favor of the alternative hypothesis.

Significance Level

The significance level, denoted as alpha (?), is the threshold for rejecting the null hypothesis in hypothesis testing. It represents the probability of making a Type I error—rejecting the null hypothesis when it is actually true. The chosen alpha level sets the stringency of the test criteria.

t-Distribution

The t-distribution is a probability distribution that is symmetric and bell-shaped, similar to the normal distribution but with heavier tails. It is used in situations involving small sample sizes or when the population variance is unknown, as it provides a more accurate reflection of uncertainty in the estimate of the standard deviation.

One-Sample t-Test

The one-sample t-test is used to compare the mean of a single sample to a known or hypothesized population mean when the population standard deviation is unknown. It relies on the t-distribution, which is more appropriate than the normal distribution for small sample sizes because it accounts for increased uncertainty in the estimation of the sample standard deviation.

Hypothesis Formulation

In hypothesis testing, one begins by formulating a null hypothesis (H0) and an alternative hypothesis (H1). The null hypothesis generally represents the status quo or a statement of no effect—in this case, that the true average study time meets or exceeds a specified value—while the alternative hypothesis represents the claim being tested, typically challenging the null with a direction (e.g., less than the stated value).

Transcript

00:01 This is problem 76 out of section 93.

00:05 This problem is about first year college students and their studying habits.

00:10 So the university claims that the students study two and a half hours per week, which we're going to change in the minutes here.

00:17 It's going to be 150 minutes.

00:20 Someone's skeptical about that and thinks they actually study less.

00:24 They took a random sample of 30 first year students and they got some information here.

00:29 The sample of the 30 students said that they study 137 minutes with a sample standard deviation of 45 minutes.

00:42 Remember, the sample size was 30.

00:45 We look like we're in a mean here, so we're going to need degrees of freedom.

00:49 Degrees of freedom is a sample size minus 1, so degrees of freedom is 29.

00:54 And they want to know is their convincing evidence that they studied less than this.

01:00 So this is going to be a state plan to conclude.

01:04 Our statement, we will perform a one sample t test, one sample t test, at an alpha value of 0 .055%, to see if freshmen really study less than 150.

01:29 Minutes per night.

01:32 We do need a null hypothesis and an alternate hypothesis.

01:37 My null hypothesis is the claim from the college, so mu equals 150 minutes.

01:44 My alternate hypothesis is what i'm trying to figure out.

01:49 Mu is less than 150 minutes.

01:53 After the statement, we move on to our plan.

01:56 Remember three things in the plan.

01:57 And random, 10 % rule, large counts.

02:01 Oh, random.

02:03 Looking at the problem, they randomly selected 30 first year students, so we're good.

02:08 10 % rule, i am going to assume that 30 is less than 10 % of all first year college students.

02:15 That means there's more than 300 first year college students that will say yes.