Test the appropriate hypotheses. Use the p-value method here. Annual per capita consumption of milk is 21.6 gallons. A sample of 16 individuals from the midwestern town of Webster City showed a sample mean annual consumption of 24.1 gallons with a standard deviation of s=4.8. Develop a hypothesis test that can be used to determine whether the mean annual consumption in Webster City equals to the national mean. Hint: Notice that n=16<30 hence you need to use the t-table. Hence, locate the probability for one (right) tail. You won't get one single value, but a range of possible values. Then multiply it by two (to take into account the left tail) and compare to 0.05. a. What is a point estimate of the difference between mean annual consumption in Webster City and the national mean? 2.083 b. At a =.05, test for a significant difference. What is your conclusion? Reject ¶
Test the appropriate hypotheses. Use the p-value method here. Annual per capita consumption of milk is 21.6 gallons. A sample of 16 individuals from the midwestern town of Webster City showed a sample mean annual consumption of 24.1 gallons with a standard deviation of s=4.8. Develop a hypothesis test that can be used to determine whether the mean annual consumption in Webster City equals to the national mean. Hint: Notice that n=16<30 hence you need to use the t-table. Hence, locate the probability for one (right) tail. You won't get one single value, but a range of possible values. Then multiply it by two (to take into account the left tail) and compare to 0.05. a. What is a point estimate of the difference between mean annual consumption in Webster City and the national mean? 2.083 b. At a =.05, test for a significant difference. What is your conclusion? Reject ¶
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 Using the p-Value Method**
**Scenario:**
The annual per capita consumption of milk is reported to be 21.6 gallons nationwide. A sample of 16 individuals from the midwestern town of Webster City reveals a sample mean annual consumption of 24.1 gallons with a standard deviation of 4.8. The goal is to develop a hypothesis test to determine if the mean annual consumption in Webster City equals the national mean.
**Step-by-Step Solution:**
**1. Formulate the Hypotheses:**
- **Null Hypothesis (H0):** The mean annual milk consumption in Webster City is equal to the national mean.
\[
H_0: \mu = 21.6
\]
- **Alternative Hypothesis (H1):** The mean annual milk consumption in Webster City is not equal to the national mean.
\[
H_1: \mu \neq 21.6
\]
**2. Information and Calculations:**
- Sample size (n): 16
- Sample mean (\(\bar{x}\)): 24.1
- Population mean (\(\mu\)): 21.6
- Standard deviation (s): 4.8
**Note:** Since \( n < 30 \), a t-test is appropriate, and we refer to the t-distribution.
**3. Calculate Point Estimate:**
What is the point estimate of the difference between Webster City's mean annual consumption and the national mean?
\[
\text{Point Estimate} = \bar{x} - \mu = 24.1 - 21.6 = 2.083
\]
**4. Test the Significance:**
Using α = 0.05, we need to locate the critical value from the t-distribution table and compute the test statistic. Given that \( n=16 \), degrees of freedom (df) are \( n-1 = 15 \).
**a. Compute the test statistic (t):**
\[
t = \frac{\bar{x} - \mu}{s / \sqrt{n}} = \frac{24.1 - 21.6}{4.8 / \sqrt{16}} = \frac{2.5}{1.2} \approx 2.083
\]
**b. Determine the p-value:**
Using t-distribution tables or software for \( df = 15](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffe12f3ad-f419-427a-9045-294545f74979%2Faa82b3e0-b408-44bd-8c4d-4029ac19a3a3%2Fv10skg_processed.png&w=3840&q=75)
Transcribed Image Text:**Hypothesis Testing Using the p-Value Method**
**Scenario:**
The annual per capita consumption of milk is reported to be 21.6 gallons nationwide. A sample of 16 individuals from the midwestern town of Webster City reveals a sample mean annual consumption of 24.1 gallons with a standard deviation of 4.8. The goal is to develop a hypothesis test to determine if the mean annual consumption in Webster City equals the national mean.
**Step-by-Step Solution:**
**1. Formulate the Hypotheses:**
- **Null Hypothesis (H0):** The mean annual milk consumption in Webster City is equal to the national mean.
\[
H_0: \mu = 21.6
\]
- **Alternative Hypothesis (H1):** The mean annual milk consumption in Webster City is not equal to the national mean.
\[
H_1: \mu \neq 21.6
\]
**2. Information and Calculations:**
- Sample size (n): 16
- Sample mean (\(\bar{x}\)): 24.1
- Population mean (\(\mu\)): 21.6
- Standard deviation (s): 4.8
**Note:** Since \( n < 30 \), a t-test is appropriate, and we refer to the t-distribution.
**3. Calculate Point Estimate:**
What is the point estimate of the difference between Webster City's mean annual consumption and the national mean?
\[
\text{Point Estimate} = \bar{x} - \mu = 24.1 - 21.6 = 2.083
\]
**4. Test the Significance:**
Using α = 0.05, we need to locate the critical value from the t-distribution table and compute the test statistic. Given that \( n=16 \), degrees of freedom (df) are \( n-1 = 15 \).
**a. Compute the test statistic (t):**
\[
t = \frac{\bar{x} - \mu}{s / \sqrt{n}} = \frac{24.1 - 21.6}{4.8 / \sqrt{16}} = \frac{2.5}{1.2} \approx 2.083
\]
**b. Determine the p-value:**
Using t-distribution tables or software for \( df = 15
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