Independent random samples of business managers and college...

Independent random samples of business managers and college economics faculty were asked to respond on a scale from 1 (strongly disagree) to 7 (strongly agree) to this statement: Grades in advanced economics are good indicators of students' analytical skills. For a sample of 70 business managers, the mean response was 4.4 and the sample standard deviation was 1.3. For a sample of 106 economics faculty the mean response was 5.3 and the sample standard deviation was 1.4. a. Test, at the $5 \%$ level, the null hypothesis that the population mean response for business managers would be at most 4.0 . b. Test, at the $5 \%$ level, the null hypothesis that the population means are equal against the alternative that the population mean response is higher for economics faculty than for business managers.

Independent random samples of business managers and college economics faculty were asked to respond on a scale from 1 (strongly disagree) to 7 (strongly agree) to this statement: Grades in advanced economics are good indicators of students' analytical skills. For a sample of 70 business managers, the mean response was 4.4 and the sample standard deviation was 1.3. For a sample of 106 economics faculty the mean response was 5.3 and the sample standard deviation was 1.4.
a. Test, at the $5 \%$ level, the null hypothesis that the population mean response for business managers would be at most 4.0 .
b. Test, at the $5 \%$ level, the null hypothesis that the population means are equal against the alternative that the population mean response is higher for economics faculty than for business managers.

Key Concepts

Significance Level

The significance level, typically denoted by alpha and often set at 0.05, is the threshold probability used to decide when to reject the null hypothesis. It represents the maximum risk of committing a Type I error—the error of rejecting a true null hypothesis—that the researcher is willing to accept.

t-Distribution and t-Test

The t-distribution is a probability distribution used in situations where the sample size is small or the population variance is unknown. The t-test employs this distribution to test hypotheses about population means by calculating a test statistic from the sample data and comparing it to a critical value from the t-distribution, thereby determining the likelihood of observing the data if the null hypothesis were true.

Hypothesis Testing

Hypothesis testing is a fundamental method in statistics used to make inferences about populations based on sample data. It involves setting up two competing hypotheses—the null hypothesis, which represents a default or no-effect state, and the alternative hypothesis, which represents the presence of an effect or difference—and then using a test statistic to determine whether to reject the null hypothesis in favor of the alternative.

Null and Alternative Hypotheses

The null hypothesis (H0) is the statement of no difference or no effect, serving as the starting assumption for any statistical test, while the alternative hypothesis (Ha) represents the claim for which evidence is sought, indicating a difference or effect. Proper formulation of these hypotheses is crucial for guiding the subsequent choice of test and the interpretation of results.

One-Sample and Two-Sample t-Tests

A one-sample t-test is used to determine whether the mean of a single sample is significantly different from a known or hypothesized population mean, while a two-sample t-test compares the means of two independent samples to assess whether there is a statistically significant difference between them. Both tests account for the variability within the data and are based on assumptions about the underlying distribution of the populations.

Transcript

00:01 So we have a group of business managers and has a group of 70 with a mean and a standard deviation and a group of economics faculty, 106 with a mean and a standard deviation for whatever that scale was.

00:15 And we want to find in part a, do we have evidence that this group has a mean that is higher than four? they claim that the mean is at most four, so less than or equal to four, and alternately that that mean is higher than four.

00:34 So we want to find out at a 5 % significance level, do we have evidence to support, to reject the null, or whether we don't have information to reject the null? so we know that we're going to be using a t test.

00:48 We're assuming that our mean of our sampling distribution is centered at four, but we're getting.

00:54 Of 4 .4.

00:56 So what's the likelihood of getting an x bar greater than a equal to 4 .4 if the sample mean is actually 4.

01:04 So let's convert that to a t value with 69 degrees of freedom.

01:09 And that becomes 4 .4 minus 4 divided by the standard deviation over the square root of n.

01:16 And that test statistic comes out to be, comes out to be 2 .574.

01:25 And this is a number.

01:26 The p value or the likelihood of getting that value or higher with that test statistic, using my software, becomes .006.

01:35 So this is much smaller, much less than the 5 % significance level.

01:42 Therefore, we have evidence to reject the null.

01:47 So the claim that they believe that they're at most for, we would reject that.

01:53 And appears as though they're that mean of the business managers, that that is actually greater than four is what we would conclude.

02:02 Now on part b, we want to find out, do we have evidence whether the economics faculty mean is equal to or the difference between that and the business manager is zero? or whether that difference is greater than zero, meaning that the engineering or the economic faculty have a higher mean score.

02:30 And again, we're going to use that 5 % significance level.

02:34 So our test statistic is going to be a t value.

02:38 And when i go and find what that value is going to be, it ends up that our test statistic, when we've determined what the degrees of freedom is, we would use if we're going to be conservative, the one less than the smallest sample size.

02:57 And otherwise, let me quick get these pieces of information in.

03:04 Let's see.

03:06 I happen to be putting this information just to shorten my work as far as doing the calculation as a two -sample t test, and we are not pooling.

03:18 So i'm doing this sample and we are not pooling.

03:25 And in any case, we would be calculating our test statistic.

03:30 And as i said, we would use conservatively, we would 69 degrees of freedom.

03:34 However, if we utilize the software, the software would tell us that the degrees of freedom is...

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