The owner of Bun 'N' Run Hamburger wishes to compare the...

The owner of Bun 'N' Run Hamburger wishes to compare the sales per day at two locations. The mean number sold for 10 randomly selected days at the Northside site was 83.55, and the standard deviation was $10.50 .$ For a random sample of 12 days at the Southside location, the mean number sold was 78.80 and the standard deviation was $14.25 .$ At the .05 significance level, is there a difference in the mean number of hamburgers sold at the two locations? What is the $p$ -value?

Key Concepts

Degrees of Freedom

Degrees of freedom refer to the number of independent values that can vary in an analysis without breaking any constraints. In the context of the two-sample t-test, especially when assuming unequal variances, the degrees of freedom are approximated using a formula such as the Welch-Satterthwaite equation. This approximation affects the critical values and ultimately the p-value in the test.

Significance Level

The significance level, often represented by alpha (?), is a threshold used in hypothesis testing to decide whether to reject the null hypothesis. Commonly set at 0.05, it indicates the probability of rejecting the null hypothesis when it is actually true. Results with p-values less than the significance level are generally considered statistically significant.

P-value

The p-value is a measure that helps determine the significance of the test results. It represents the probability of obtaining a test statistic at least as extreme as the one observed, under the assumption that the null hypothesis is true. A smaller p-value indicates stronger evidence against the null hypothesis, leading to its rejection if the p-value falls below the predetermined significance level.

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 holds for the entire population. In the context of comparing two means, the null hypothesis typically states that there is no difference between the population means, while the alternative hypothesis suggests that a difference exists.

Two?Sample t-Test

The two-sample t-test is a statistical procedure used to compare the means of two independent groups to see if they are statistically significantly different from each other. This test is especially useful when the sample sizes are small and the population variances are unknown, often requiring adjustments for unequal variances through methods like the Welch-Satterthwaite approximation.

Transcript

00:01 So we want to determine whether the mean sales at the north side store is equal to that of the south side store and alternately that they're different.

00:11 So we're doing a two -tail test.

00:13 And we know that we have from the north side, we have the sample size is 10 with a mean of 83 .55 and a standard deviation of 10 .50.

00:28 And from the north, excuse me, the south side, we have a sample of 12 that yielded a mean of 78 .8 and a standard deviation of 14 .25.

00:43 So we're going to assume that these two distributions are approximately normal, and we're also going to assume that these standard deviations are approximately the same.

00:52 And we need to find our pooled variance.

00:56 And our pooled variance is going to be to take the first sample size less 1 times the variance of the first group, which is going to be that 10 .5 squared.

01:08 And then one less than the sample size of the second group times the variance of the second group.

01:14 And that is a 25.

01:16 And then we'll divide that by that 10 plus 12 less 2.

01:21 And so that's going to end up being 20.

01:23 And our degrees of freedom will end up being 20 for this setting 2.

01:28 And our two sample t test.

01:30 So let me quick get that variance for you.

01:33 And we have left parentheses 9 times 10 .5 squared plus 11 times 14 .25 squared.

01:43 Close my parentheses off and then divide that by 20.

01:46 And i get that that variance, pulled variance, is a hundred thousand.

01:51 161 .297.

01:54 Now i'm going to actually store that value in my calculator as x.

01:58 And now we need to calculate our test statistic.

02:01 Oh, and let me see what if we had a specific level.

02:06 It was a 5 % significance level.

02:08 So we can do this.

02:10 We can come up and find out what our critical values are for our two -sided test and when i look at my book for 20 degrees of freedom and we would want for a 5 % significance test significance level we are going to put 0 .025 at each tail so half of the variability at each tail and this t value that will have 20 degrees of freedom and has 0 .025 in the upper tail is 2 .086 2 .086 and this one is negative 2 .086 and this one is negative 2 .08 and so we would reject the null, reject the null if we are below this lower value or higher than the other value, and we would fail to reject in between.

02:59 So let's calculate our test statistic now...

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