The Grand Strand Family Medical Center is specifically set...

The Grand Strand Family Medical Center is specifically set up to treat minor medical emergencies for visitors to the Myrtle Beach area. There are two facilities, one in the Little River Area and the other in Murrells Inlet. The Quality Assurance Department wishes to compare the mean waiting time for patients at the two locations. Samples of the waiting times, reported in minutes, follow: At the .05 significance level, is there a difference in the mean waiting time?

Key Concepts

Hypothesis Testing

Hypothesis testing is a fundamental process in inferential statistics that involves making decisions using sample data. It starts with a null hypothesis, which usually represents a statement of no effect or no difference, and an alternative hypothesis which represents the presence of an effect or difference. The goal is to determine whether the observed data provide sufficient evidence to reject the null hypothesis in favor of the alternative.

Two-Sample t-Test

The two-sample t-test is a statistical method used to compare the means of two independent groups. It assumes that the samples are drawn from normally distributed populations (or the sample sizes are large enough for the Central Limit Theorem to apply) and helps determine if the difference in sample means is statistically significant or likely due to random chance.

Significance Level

The significance level, denoted by alpha (?), is the probability threshold used to determine whether the observed data are sufficiently unlikely under the null hypothesis. Commonly set at 0.05, it represents a 5% risk of falsely rejecting the null hypothesis and plays a critical role in decision making during hypothesis testing.

Null and Alternative Hypotheses

The null hypothesis (H0) typically indicates that there is no difference or effect, such as stating that the mean waiting times are equal across the two groups. The alternative hypothesis (H1) suggests that there is a difference or some effect, such as a difference in mean waiting times. These hypotheses form the basis for testing and deciding whether any observed differences are statistically meaningful.

p-Value

The p-value is the probability of obtaining a test statistic at least as extreme as the one observed, assuming that the null hypothesis is true. It quantifies the strength of the evidence against the null hypothesis; a smaller p-value indicates stronger evidence in favor of the alternative hypothesis. In hypothesis testing, if the p-value is less than the significance level, the null hypothesis is rejected.

Transcript

00:01 So we want to determine at a 5 % significance level, while the mean waiting time is different for these two.

00:06 And i'm just going to have to subscript them as one and two.

00:12 So find out if there's a difference.

00:15 And first of all, i found it kind of odd that they had the waiting times out to the nearest hundredth of a minute.

00:21 But in any case, i thought that was kind of interesting.

00:24 So our first sample was a sample size of 10.

00:28 The second one was of 12.

00:32 And the first sample, the mean waiting time for that first sample, was 27 .461.

00:40 And the second sample size had a mean waiting time of 25 .689 minutes.

00:46 And the two sample standard deviations, the first one was 4 .4 .0.

00:54 And the second one, in nuts and minutes, 2 .68.

00:57 And it will go 4 .5.

00:59 I don't want to do that rounding up to five.

01:02 And these are in minutes.

01:03 Now, this technique, we will be assuming that these two standard deviations are equal, and that our populations are approximately normal, and we will have to do a two sample t test.

01:15 We can't use z values.

01:18 Our sample sizes are not big enough, and we don't know that the populations are normal.

01:22 So we will do a two sample t test, and that's going to call for us to end up finding our pooled standard deviation, or our pooled variance, actually.

01:32 And we know that our degrees of freedom will end up being the 10 plus 12 less 2.

01:39 So it's going to be 20.

01:40 And why don't we just draw out before we calculate that pool variance where we will be doing our rejecting the null.

01:49 And since we're using a 5 % significance level, we'll have half of that at the upper tail, half of that at the lower tail, and we want a t value that has 20 degrees of free...

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