According to a report published by the USDA 2 years ago, a typical American consumes an average of 31.1 pounds of cheese per year. A researcher at the USDA would like to determine if the average amount of cheese consumed each year by a typical American has changed since the original report was published. The researcher collected data from a random sample of 64 Americans and found that the mean amount of cheese consumed each year by the sample was 31.65 pounds with a standard deviation of 2.915 pounds. Using a significance level of 0.1%, test the hypothesis that the mean amount of cheese consumed each year by a typical American is different than 31.1 pounds. Use the p- value method. State the null and alternative hypothesis for this test. Ho: ? ◊ H₁: ? î Determine if this test is left-tailed, right-tailed, or two- tailed. two-tailed right-tailed left-tailed Should the standard normal (z) distribution or Student's (t) distribution be used for this test? The standard normal (z) distribution should be used The Student's t distribution should be used Determine the test statistic for the hypothesis test. Round the solution to four decimal places. Determine the p-value (range) for the hypothesis test.
According to a report published by the USDA 2 years ago, a typical American consumes an average of 31.1 pounds of cheese per year. A researcher at the USDA would like to determine if the average amount of cheese consumed each year by a typical American has changed since the original report was published. The researcher collected data from a random sample of 64 Americans and found that the mean amount of cheese consumed each year by the sample was 31.65 pounds with a standard deviation of 2.915 pounds. Using a significance level of 0.1%, test the hypothesis that the mean amount of cheese consumed each year by a typical American is different than 31.1 pounds. Use the p- value method. State the null and alternative hypothesis for this test. Ho: ? ◊ H₁: ? î Determine if this test is left-tailed, right-tailed, or two- tailed. two-tailed right-tailed left-tailed Should the standard normal (z) distribution or Student's (t) distribution be used for this test? The standard normal (z) distribution should be used The Student's t distribution should be used Determine the test statistic for the hypothesis test. Round the solution to four decimal places. Determine the p-value (range) for the hypothesis test.
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![### Hypothesis Testing Example: Cheese Consumption in America
**Background:**
According to a report published by the USDA two years ago, a typical American consumes an average of 31.1 pounds of cheese per year.
**Research Context:**
A researcher at the USDA would like to determine if the average amount of cheese consumed each year by a typical American has changed since the original report was published. The researcher collected data from a random sample of 64 Americans and found that the mean amount of cheese consumed each year by the sample was 31.65 pounds with a standard deviation of 2.915 pounds.
**Objective:**
Using a significance level of 0.1%, test the hypothesis that the mean amount of cheese consumed each year by a typical American is different from 31.1 pounds. Use the p-value method.
---
### Steps to Conduct the Hypothesis Test
**1. State the Null and Alternative Hypotheses:**
- \( H_0 \): The null hypothesis is that the mean amount of cheese consumed each year by a typical American is 31.1 pounds.
- \( H_1 \): The alternative hypothesis is that the mean amount of cheese consumed each year by a typical American is not 31.1 pounds.
**2. Determine if the Test is Left-Tailed, Right-Tailed, or Two-Tailed:**
Given that the alternative hypothesis is that the mean amount of cheese consumed is different from 31.1 pounds, this is a two-tailed test.
- ☐ Two-tailed
- ☐ Right-tailed
- ☐ Left-tailed
Select: Two-tailed
**3. Choose the Appropriate Distribution:**
- ☐ The standard normal (\( z \)) distribution should be used
- ☐ The Student's \( t \)) distribution should be used
Given the sample size (n = 64) and unknown population standard deviation, the Student's \( t \)) distribution should be used.
**4. Calculate the Test Statistic:**
To conduct the test using the Student’s \( t \) distribution, compute the test statistic using the following formula:
\[ t = \frac{\bar{x} - \mu}{(s / \sqrt{n})} \]
Where:
- \( \bar{x} = 31.65 \)
- \( \mu = 31.1 \)
- \( s = 2.915 \)
- \( n = 64 \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faf14cf2c-4c6b-44b0-8c19-b82da3615fd0%2F0ff43bf0-4142-42cf-9059-01d7f9d5099b%2Fflnkwij_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Hypothesis Testing Example: Cheese Consumption in America
**Background:**
According to a report published by the USDA two years ago, a typical American consumes an average of 31.1 pounds of cheese per year.
**Research Context:**
A researcher at the USDA would like to determine if the average amount of cheese consumed each year by a typical American has changed since the original report was published. The researcher collected data from a random sample of 64 Americans and found that the mean amount of cheese consumed each year by the sample was 31.65 pounds with a standard deviation of 2.915 pounds.
**Objective:**
Using a significance level of 0.1%, test the hypothesis that the mean amount of cheese consumed each year by a typical American is different from 31.1 pounds. Use the p-value method.
---
### Steps to Conduct the Hypothesis Test
**1. State the Null and Alternative Hypotheses:**
- \( H_0 \): The null hypothesis is that the mean amount of cheese consumed each year by a typical American is 31.1 pounds.
- \( H_1 \): The alternative hypothesis is that the mean amount of cheese consumed each year by a typical American is not 31.1 pounds.
**2. Determine if the Test is Left-Tailed, Right-Tailed, or Two-Tailed:**
Given that the alternative hypothesis is that the mean amount of cheese consumed is different from 31.1 pounds, this is a two-tailed test.
- ☐ Two-tailed
- ☐ Right-tailed
- ☐ Left-tailed
Select: Two-tailed
**3. Choose the Appropriate Distribution:**
- ☐ The standard normal (\( z \)) distribution should be used
- ☐ The Student's \( t \)) distribution should be used
Given the sample size (n = 64) and unknown population standard deviation, the Student's \( t \)) distribution should be used.
**4. Calculate the Test Statistic:**
To conduct the test using the Student’s \( t \) distribution, compute the test statistic using the following formula:
\[ t = \frac{\bar{x} - \mu}{(s / \sqrt{n})} \]
Where:
- \( \bar{x} = 31.65 \)
- \( \mu = 31.1 \)
- \( s = 2.915 \)
- \( n = 64 \
Transcribed Image Text:### Determining the p-Value and Drawing Conclusions for Hypothesis Tests
#### Step 1: Determine the p-value (range) for the hypothesis test.
- **Select the appropriate p-value range:**
- ⬜ p-value < 0.001
- ⬜ 0.001 < p-value < 0.01
- ⬜ 0.01 < p-value < 0.02
- ⬜ 0.02 < p-value < 0.05
- ⬜ 0.05 < p-value < 0.10
- ⬜ 0.10 < p-value < 0.20
- ⬜ p-value > 0.20
#### Step 2: Determine the appropriate conclusion for this hypothesis test.
- **Select the conclusion that best fits the data:**
- ⬜ The sample data do not provide sufficient evidence to reject the null hypothesis that the mean amount of cheese consumed each year by a typical American is 31.1 pounds and thus we conclude that the mean amount of cheese consumed each year by a typical American is likely 31.1 pounds.
- ⬜ The sample data provide sufficient evidence to reject the alternative hypothesis that the mean amount of cheese consumed each year by a typical American is different than 31.1 pounds and thus we conclude that the mean amount of cheese consumed each year by the typical American is likely 31.1 pounds.
- ⬜ The sample data provide sufficient evidence to reject the null hypothesis that the mean amount of cheese consumed each year by a typical American is 31.1 pounds and thus we conclude that the mean amount of cheese consumed each year by a typical American is likely different than 31.1 pounds.
- ⬜ The sample data do not provide sufficient evidence to reject the alternative hypothesis that the mean amount of cheese consumed each year by a typical American is different than 31.1 pounds and thus we conclude that the mean amount of cheese consumed each year by the typical American is likely different than 31.1 pounds.
#### Explanation of Graphs or Diagrams
This section would normally include visual aids such as charts or graphs to support the p-value and hypothesis test conclusions. Diagrams commonly used include:
- **Histograms:** Showing distribution of cheese consumption data.
- **Box Plots:** Illustrating the spread and central tendency of the
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