(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. At the time she was hired as a server at the Grumney Family Restaurant, Beth Brigden was told, "You can average $\$ 80$ a day in tips." Assume the population of daily tips is normally distributed with a standard deviation of $\$ 9.95 .$ Over the first 35 days she was employed at the restaurant, the mean daily amount of her tips was \$84.85. At the .01 significance level, can Ms. Brigden conclude that her daily tips average more than $\$ 80 ?$

(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.
At the time she was hired as a server at the Grumney Family Restaurant, Beth Brigden was told, "You can average $\$ 80$ a day in tips." Assume the population of daily tips is normally distributed with a standard deviation of $\$ 9.95 .$ Over the first 35 days she was employed at the restaurant, the mean daily amount of her tips was \$84.85. At the .01 significance level, can Ms. Brigden conclude that her daily tips average more than $\$ 80 ?$

Key Concepts

Null and Alternate Hypotheses

In hypothesis testing, the null hypothesis (H?) is a statement of no effect or no difference, while the alternate (or alternative) hypothesis (H?) posits a specific effect or difference that the test seeks to detect. These hypotheses form the basis of the statistical test, and the formulation explicitly defines whether the test is one-tailed or two-tailed.

Decision Rule

The decision rule in hypothesis testing determines the criteria under which the null hypothesis is rejected or not. It involves comparing the computed test statistic against a critical value derived from the chosen significance level and the distribution of the test statistic. If the statistic falls in the rejection region, H? is rejected.

Test Statistic

The test statistic quantifies the difference between the sample statistic and the hypothesized population parameter, standardized by the variability in the data. When the population standard deviation is known and the sample comes from a normally distributed population, the z-test is typically used, with the test statistic computed as (sample mean - hypothesized mean) divided by (population standard deviation divided by the square root of the sample size).

p-value

The p-value represents the probability of obtaining a test statistic at least as extreme as the one computed from the sample data assuming that the null hypothesis is true. It provides a measure of the strength of the evidence against H?. A smaller p-value indicates stronger evidence in favor of the alternative hypothesis.

Significance Level

The significance level, often denoted by ?, is the threshold used to decide whether to reject the null hypothesis. It represents the probability of making a Type I error, which is rejecting H? when it is actually true. The decision rule compares the p-value or test statistic to the critical value corresponding to this ? level.

Transcript

00:02 All right.

00:04 So we have a server who's told they can average 80 bucks and tips a day.

00:13 Assuming this population of tips is normally distributed with a given standard deviation, we want to know if the daily amount of tips you received, $84 .25, is significant.

00:31 And let's make sure we ask, we're looking at the right question.

00:35 So the sample size is 35 days and the average day, the mean daily amount of her tips was that $84 .85.

00:43 And can the server conclude that her daily tips average more than $80? so assuming that this $80 is the population, we don't know if it's greater than, if this $84 is significantly, $84 is significantly greater than so we're going to have the no hypothesis be the mu is less than or equal to 80.

01:11 The alternative is that no it's bigger than 80.

01:14 Like hers is a this is pretty pretty rare.

01:20 That's what we're looking at here.

01:22 So let's get our critical value.

01:32 Like if it does she fall within the population? that's what this is.

01:36 So let's take a look at that.

01:40 So we want the point -on -one significance level.

01:43 This is a function that gives me the critical region.

01:47 And it's a one -tailed test.

01:51 Oh, we actually want my 9 because that's the positive value.

01:58 That's on the right of our curve.

02:02 And so the decision rule is going to be if our z to z statistic greater than this critical value...