(a) State the null hypothesis and the alternate hypothesis....

(a) State the null hypothesis and the alternate hypothesis. (b) State the decision rule. (c) Compute the value of the test statistic. (d) What is your decision regarding $H_{0} ?$ (e) What is the $p$ -value? Interpret it. The waiting time for customers at MacBurger Restaurants follows a normal distribution with a population standard deviation of 1 minute. At the Warren Road MacBurger, the quality assurance department sampled 50 customers and found that the mean waiting time was 2.75 minutes. At the . 05 significance level, can we conclude that the mean waiting time is less than 3 minutes?

(a) State the null hypothesis and the alternate hypothesis. (b) State the decision rule. (c) Compute the value of the test statistic. (d) What is your decision regarding $H_{0} ?$ (e) What is the $p$ -value? Interpret it.
The waiting time for customers at MacBurger Restaurants follows a normal distribution with a population standard deviation of 1 minute. At the Warren Road MacBurger, the quality assurance department sampled 50 customers and found that the mean waiting time was 2.75 minutes. At the . 05 significance level, can we conclude that the mean waiting time is less than 3 minutes?

Key Concepts

Null and Alternative Hypotheses

This concept involves formulating two competing statements about a population parameter. The null hypothesis (H?) generally asserts that no effect or no difference exists (for example, that a parameter equals a specific value), while the alternative hypothesis (H?) proposes that there is an effect or a difference (such as the parameter being less than, greater than, or not equal to the specified value). This formulation sets the foundation for conducting a statistical test.

Significance Level and Decision Rule

The significance level (?) is a predetermined threshold from the probability domain (commonly 0.05) that represents the maximum risk of committing a Type I error, which is rejecting the null hypothesis when it is actually true. The decision rule is derived from this significance level by identifying critical values from the relevant probability distribution. If the test statistic falls in the critical region, the null hypothesis is rejected; otherwise, it is not rejected.

Test Statistic

A test statistic is a standardized value computed from the sample data that measures the degree of divergence from what is expected under the null hypothesis. In this context, when the population standard deviation is known and the sample mean is compared to a hypothesized mean, a z-test is typically used. The formula involves subtracting the hypothesized mean from the sample mean and dividing the result by the standard error of the mean.

p-value

The p-value quantifies 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 the evidence against the null hypothesis: a smaller p-value indicates stronger evidence in favor of the alternative hypothesis. If the p-value is less than the chosen significance level, it suggests that the observed result is statistically significant and grounds are sufficient for rejecting the null hypothesis.

Transcript

00:01 All right, so we have a restaurant.

00:04 The waiting time at the restaurant follows a normal distribution, the population standardvation of the minute, and at a specific one, specific of these restaurants, they sampled 50 customers and found the waiting time was 2 .75.

00:19 And at the 0 .05 significance level, can we conclude that the mean waiting time is less than three minutes? so is this observed? reasonable.

00:34 So the alternative, the null hypothesis is that the mean, well, i'm just going to write you, that it should be new.

00:45 What you want to see is it greater than equal to three minutes, but the hypothesis that the mean, is that the mean is less than three.

01:00 That's what it says here.

01:01 Can we conclude that the mean waiting time is less than three minutes? we're assuming it's three or greater.

01:06 So we're going to see, hey, is it truly less than three minutes? so the decision rule, we're looking for the z score that would correspond with 0 .05.

01:26 And it's this.

01:28 So this is what we want, negative 1 .6, because is it less than is it negative? that's what we want.

01:34 So is the z value less than this critical value? that's what we're looking for.

01:40 So let's do the calculation here.

01:43 So the sample mean is 2 .75 and then we do minus three.

01:48 It's the sample.

01:49 It's the assume population and the population mean...

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