Refer to the Lincolnville School District bus data. a....

Refer to the Lincolnville School District bus data. a. Select the variable for the number of miles traveled last month. Conduct a hypothesis test to determine whether the mean miles traveled last month equals $10,000 .$ Use the .01 significance level. Find the $p$ -value and explain what it means. b. A study of school bus fleets reports that the average maintenance cost per bus is $\$ 4,000$ per year. Using the maintenance cost variable, conduct a hypothesis test to determine whether the mean maintenance cost for Lincolnville's bus fleet is more than $\$ 4,000$ at the .05 significance level. Determine the $p$ -value and report the results.

Refer to the Lincolnville School District bus data.
a. Select the variable for the number of miles traveled last month. Conduct a hypothesis test to determine whether the mean miles traveled last month equals $10,000 .$ Use the .01 significance level. Find the $p$ -value and explain what it means.
b. A study of school bus fleets reports that the average maintenance cost per bus is $\$ 4,000$ per year. Using the maintenance cost variable, conduct a hypothesis test to determine whether the mean maintenance cost for Lincolnville's bus fleet is more than $\$ 4,000$ at the .05 significance level. Determine the $p$ -value and report the results.

Key Concepts

p-value

The p-value is the probability, under the assumption that the null hypothesis is true, of obtaining a test statistic at least as extreme as the one computed from the sample data. It quantifies the evidence against the null hypothesis; a smaller p-value indicates stronger evidence in favor of the alternative hypothesis.

Significance Level

The significance level, denoted by alpha (?), is the threshold for determining whether the observed data is statistically significant. It represents the probability of rejecting the null hypothesis when it is actually true. Common significance levels include 0.05 and 0.01, and they dictate how strong the evidence must be to reject the null hypothesis.

One-tailed and Two-tailed Tests

One-tailed tests are used when the research hypothesis specifies a direction of the effect (for example, greater than a certain value), while two-tailed tests are used when the direction is not specified, allowing for deviations in both directions. The choice between one-tailed and two-tailed tests affects how the p-value is interpreted and how the critical regions are determined.

Hypothesis Testing

Hypothesis testing is a statistical method that uses sample data to evaluate a hypothesis about a population parameter. This process involves setting up a null hypothesis, which typically represents a state of no effect or a specified value, and an alternative hypothesis, which proposes a deviation from that value. The outcome is determined by comparing a calculated test statistic to a critical value or by evaluating the p-value against a preset significance level.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis (H?) represents the assumed equality or status quo (such as a mean value being equal to a specific number), while the alternative hypothesis (H?) represents the claim that there is a significant difference or effect (for example, the mean being either different from or greater than the specific number). These form the foundation of the test and guide the direction of the analysis.

Transcript

00:01 All right, so we're looking at our bus data, and we're going to make a, try and determine at the 0 .01 significance level, if the mean number of miles traveled last month is 10 ,000.

00:16 So there's our hypothesis that the assumed mean is 10 ,000.

00:22 So the alternative is that it's not equal to 10 ,000.

00:25 So this is the mean for the miles driven last month.

00:29 It's standard deviation.

00:30 It's a t statistics since it's a sample.

00:33 And so we need a critical value.

00:37 So let's see, it's a two -tailed test at the 0 .01 level of significance, a point 0 .01 level, but we have to cut in half.

00:50 And there are 80 buses, so 79 degrees of freedom, 2 .64.

01:01 There's the critical value.

01:05 And the t value, i'm going to do this now with the mean, sample main, minus the bus room, standard deviation, squared sample size, 16, yeah, that's 16 .26, which is greater than the critical value.

01:32 Which, oh, i should have said, this is actually also, it could be less than the negative, the negative of the critical value, because it's a two -tailed test.

01:43 So that's fine.

01:46 And it's all good.

01:53 And so the p value is going to be, it's so close to zero.

01:58 I mean, we'll put a few zero there.

02:01 Lots of one there somewhere.

02:02 I don't know.

02:03 It could be wrong.

02:04 I'll go a little higher.

02:05 Maybe it's not, but it's very small.

02:09 So we reject the null hypothesis and say the mean travels, the mean miles travel last month is not equal to 10 ,000...