A recent article in Vitality magazine reported that the mean...

A recent article in Vitality magazine reported that the mean amount of leisure time per week for American men is 40.0 hours. You believe this figure is too large and decide to conduct your own test. In a random sample of 60 men, you find that the mean is 37.8 hours of leisure per week and that the standard deviation of the sample is 12.2 hours. Can you conclude that the information in the article is untrue? Use the .05 significance level. Determine the $p$ -value and explain its meaning.

Key Concepts

p-Value

The p-value is the probability of obtaining a test statistic at least as extreme as the one observed, under the assumption that the null hypothesis is true. It quantifies the evidence against the null hypothesis; a smaller p-value indicates stronger evidence that the observed data is inconsistent with the null hypothesis, and thus supports the alternative hypothesis.

Significance Level

The significance level, often denoted by ?, is the threshold used to determine whether the p-value indicates a statistically significant result. It represents the probability of rejecting the null hypothesis when it is actually true (a Type I error). In hypothesis testing, comparing the p-value to the significance level helps decide if the null hypothesis should be rejected in favor of the alternative hypothesis.

Null and Alternative Hypotheses

The null hypothesis is a statement of no effect or no difference, and it serves as the starting assumption for statistical tests. In contrast, the alternative hypothesis represents a claim that contradicts the null, indicating the presence of a statistically significant difference or effect. Clearly defining these hypotheses is essential for any inferential analysis as they frame the test and the conclusions drawn from it.

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence in a sample of data to infer that a certain condition is true for the entire population. It involves formulating a null hypothesis that represents a default position and an alternative hypothesis that represents the claim being tested, followed by using sample data to determine whether the evidence is strong enough to reject the null hypothesis.

One-Sample t-Test

A one-sample t-test is used when the population standard deviation is unknown and the sample size is relatively small, making it appropriate to compare the sample mean to a known population mean. This test relies on the t-distribution, which adjusts for extra variability caused by using the sample standard deviation as an estimate of the population standard deviation.

Transcript

00:02 Can we conclude the information in this article is untrue? all right.

00:08 So let's see what this article is telling us.

00:10 So this article reported that the mean amount of leisure time per week for american men is 40 hours.

00:16 And we think this is too big.

00:19 And we decide to make our own, to conduct your own tests.

00:22 And we sample 60 men.

00:24 And we find that the mean is 37 .8 hours of leisure time per week, and with a standard deviation of 12 .2.

00:36 Can we conclude this information is untrue? so that means we're looking for is the mean not equal to this 40 meet, 40 hours, under the assumption that no, it is 40.

00:57 And i can only tell us we're going to use the t value because we have the samples.

01:02 We don't have the population standard deviation, but we do have the sample, standard deviation.

01:08 And i remember one of my old stats professors saying it's very rare that we have the population.

01:13 Standard deviation.

01:15 Sometimes we do, but a lot of times out in the field, you don't have the, that you have, you might have, it's easy to get the population mean from estimates, but it's hard to get the population standard deviation.

01:26 But that's neither here nor there.

01:27 Let's move forward.

01:27 So it's the t value regardless because we have the sample standard deviation.

01:31 So we need the critical value as well.

01:36 And we're looking at the 0 .5 significance level.

01:38 It's two -tailed, two -tail test.

01:40 So that means we need to take 0 .05 and cut it in half.

01:45 So let's see, degrees of freedom is 59, 60 minus 1.

01:49 And so it's two -tailed test at 0 .05, but this, i honestly don't like how this is broken, this table is broken down.

02:00 I just think one -tailed test, but 0 .05, you have to cut in half to make two -tailed.

02:03 So you just do that.

02:05 Then i, from my brain gets kind of confused sometimes when it has this extra stuff, then i'm already doing the calculations to confirm what's already print in front of me, but again, neither here nor there...

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