A sample of scores on an examination given in Statistics 201 are: At the .01 significance level,

A sample of scores on an examination given in Statistics 201...

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences or draw conclusions about a population parameter based on sample data. It involves formulating a null hypothesis, which represents a default position or no effect, and an alternative hypothesis, which reflects the claim of interest. The process includes calculating a test statistic and comparing it to a critical value or using a p-value to decide whether to reject the null hypothesis.

Null and Alternative Hypotheses

The null hypothesis (H?) typically states that there is no difference or effect, such as the mean grades being equal between groups. The alternative hypothesis (H?) represents the claim to be tested, which in this context is that the mean grade of one group (women) is higher than that of the other group (men). Clearly specifying these hypotheses is crucial for the testing procedure.

One-tailed Test

A one-tailed test is used when the alternative hypothesis is direction-specific; in this case, the claim is that one mean is higher than the other. This type of test focuses solely on deviations from the null hypothesis in one direction, increasing sensitivity to that specific alternative but not detecting differences in the opposite direction.

Significance Level

The significance level, often denoted as alpha (?), is the threshold used to determine whether the test statistic is extreme enough to reject the null hypothesis. In this scenario, an alpha of 0.01 indicates a 1% risk of rejecting the null hypothesis when it is actually true, thereby controlling the probability of a Type I error.

Test Statistic

The test statistic is a standardized value calculated from the sample data that is used to determine whether to reject the null hypothesis. It incorporates the difference between sample means and their variability, and its value is compared against a critical threshold from a relevant distribution, such as the t-distribution when sample standard deviations are used.

p-value

The p-value is the probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true. It provides a measure of how consistent the data is with the null hypothesis; a p-value less than the significance level (0.01 in this case) leads to the rejection of the null hypothesis.

Assumptions for the t-test

Performing a t-test to compare means requires certain assumptions: the data should be approximately normally distributed, the samples should be independent from each other, and, in some cases, the variances of the two groups should be equal or at least similar. Meeting these assumptions helps ensure the validity and reliability of the test results.

Two-sample t-test

A two-sample t-test is used to compare the means from two independent groups when the population standard deviations are unknown. It assesses whether the observed difference in sample means is statistically significant, accounting for variability in the data and the sample sizes, thus providing a framework for comparing groups such as men and women.

Transcript

00:01 So we'll be assuming that the mean for men is equal to the mean for women, and alternately that the mean for men is smaller than the mean for women.

00:11 So we're doing a one -tale test, and we're using a 1 % significance level, and we're going to have to figure out our degrees of freedom and so on because our sample sizes are small.

00:20 And if we find our mean for our first set of data, we find that our mean for the men is 78.

00:30 The mean for the women is 79.

00:34 So it doesn't look to be that much higher.

00:37 This sample standard deviation from men comes out to be 9 .48, and i'm going to call it 7.

00:44 And the mean, the sample standard deviation for women comes out to be about 6 .8, and it's going to come out to 80.

00:53 And we have our sample size for the men being nine, and we have the sample size for the women being 7.

01:01 So they're not very big sample sizes.

01:04 And we're, again, using a 1 % significance level.

01:07 And we will be calculating what that critical value is.

01:12 And in fact, why don't we just quick do that now? we know that the degrees of freedom is going to be.

01:17 We're going to be assuming here that these are equal.

01:21 We're also going to be assuming that the standard deviation of men and standard deviation of women are equal.

01:27 So we're going to need to find that pooled value.

01:29 And so we'll find this pooled variance in just one moment.

01:35 And the degrees of freedom will be the sum of these two less two.

01:39 So that's 16 less two.

01:40 That's 14.

01:41 So our t value with 14 degrees of freedom, 14 degrees of freedom.

01:47 And it's in the lower tail, so it's negative.

01:50 So 14 degrees of freedom with 1 % in that lower tail will be a negative 2 .624.

01:57 And so if we are down here, our test statistic is down here.

02:02 We'll reject the null.

02:03 Otherwise, we will fail to reject...

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