A Nissan Motor Corporation advertisement read, “The average...

Name: Problem 81, Chapter 9: Introductory Statistics
Uploaded: 2020-05-21T15:08:04-08:00
Duration: 3 min 10 s
Channel: Robin Corrigan
Description: A Nissan Motor Corporation advertisement read, “The average man’s I.Q. is 107. The average brown trout’s I.Q. is 4. So why can’t man catch brown trout?” Suppose you believe that the brown trout’s mean I.Q. is greater than four. You catch 12 brown trout. A fish psychologist determines the I.Q.s as follows: 5; 4; 7; 3; 6; 4; 5; 3; 6; 3; 8; 5. Conduct a hypothesis test of your belief.

A Nissan Motor Corporation advertisement read, “The average man’s I.Q. is 107. The average brown trout’s I.Q. is 4. So why can’t man catch brown trout?” Suppose you believe that the brown trout’s mean I.Q. is greater than four. You catch 12 brown trout. A fish psychologist determines the I.Q.s as follows: 5; 4; 7; 3; 6; 4; 5; 3; 6; 3; 8; 5. Conduct a hypothesis test of your belief.

Key Concepts

Test Statistic

The test statistic, often calculated as t = (sample mean - hypothesized mean) divided by the estimated standard error, quantifies how far the sample mean is from the hypothesized population mean. This value is compared against a t-distribution to determine the probability of observing such a difference assuming that the null hypothesis is true.

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 evidence against the null hypothesis; a smaller p-value indicates stronger evidence in favor of the alternative hypothesis, aiding in the decision-making process of whether to reject the null.

Degrees of Freedom

Degrees of freedom, calculated as the sample size minus one (n-1) in a one-sample t-test, reflect the number of independent values that can vary in the analysis while estimating statistical parameters. They are a critical component in defining the shape of the t-distribution used for the test.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis (H?) represents a statement of no effect or a baseline claim to be tested, while the alternative hypothesis (H?) represents the claim for which evidence is sought. Defining these hypotheses clearly is essential as they determine the direction and nature of the test conducted.

Hypothesis Testing

Hypothesis testing is a statistical method that allows researchers to decide, based on sample data, whether to reject a null hypothesis in favor of an alternative hypothesis. It involves comparing a test statistic computed from sample data to a distribution under the null hypothesis to assess the likelihood of observing such results by chance.

One-Sample t-Test

A one-sample t-test is used when comparing the mean of a single sample to a known or hypothesized population mean, especially when the population standard deviation is unknown and the sample size is relatively small. This test assesses whether the sample provides sufficient evidence to conclude that the population mean differs from the hypothesized value.

Transcript

00:01 We're told that it is assumed that the average brown trout's iq is 4, but that we think it is actually higher.

00:08 So from that part of the question, we know that we have an alternative hypothesis is that the mean is greater than 4.

00:18 And therefore, the null hypothesis is that the mean is at most 4.

00:27 And we're also given a sample of 12 brown trout along with their iqs.

00:32 So we have a sample size of 12.

00:34 And we're asked to conduct a hypothesis test of our belief that brown trout have an iq higher than 4.

00:40 So they don't give us a significance level to test our claim, so we can assume one ourselves.

00:45 We'll say let's do this at the significance level of 0 .05.

00:51 So from the 12 sample data that we are given in the question, you can use a calculator or a mathematical spreadsheet to calculate the sample average, and that is 4 .92.

01:05 And we're not given the population standard deviation, so we'll calculate the sample standard deviation, and that comes out to 1 .62.

01:16 So let's use the critical value approach to test our claim.

01:20 And the question of how are sample averages distributed, since we don't have the population standard deviation, but we do have the sample standard deviation, our sample averages are distributed according to the t distribution.

01:35 So using the critical value approach, we want to find, a critical value...

Please give Ace some feedback