A food manufacturer claims that eating its new cereal as part of a daily diet lowers total blood cholesterol levels. The table shows the total blood cholesterol levels (in milligrams per deciliter of blood) of seven patients before eating the cereal and after one year of eating the cereal as part of their diets. Use technology to test the mean difference. Assume the samples are random and dependent, and the population is normally distributed. At a = 0.05, can you conclude that the new cereal lowers total blood cholesterol levels? Patient 1 4 5 6 200 230 240 240 Total Blood Cholesterol (Before) 245 260 225 Total Blood Cholesterol (After) 196 227 245 236 243 255 223 Let the blood cholesterol level before eating the cereal be population 1. Let the blood cholesterol level after eating the cereal be population 2. Identify the null and alternative hypotheses, where Hd H₁-H₂. Choose the correct answer below. OA. Ho: Hd #0 OB. Ho: Hd 20 HA: Hd <0 HA: Hd=0 C. Ho: Hd ≤0 O D. Ho: Hd = 0 HA: Hd #0 HA: Hd>0 Calculate the standardized test statistic. (Round to three decimal places as needed.)
A food manufacturer claims that eating its new cereal as part of a daily diet lowers total blood cholesterol levels. The table shows the total blood cholesterol levels (in milligrams per deciliter of blood) of seven patients before eating the cereal and after one year of eating the cereal as part of their diets. Use technology to test the mean difference. Assume the samples are random and dependent, and the population is normally distributed. At a = 0.05, can you conclude that the new cereal lowers total blood cholesterol levels? Patient 1 4 5 6 200 230 240 240 Total Blood Cholesterol (Before) 245 260 225 Total Blood Cholesterol (After) 196 227 245 236 243 255 223 Let the blood cholesterol level before eating the cereal be population 1. Let the blood cholesterol level after eating the cereal be population 2. Identify the null and alternative hypotheses, where Hd H₁-H₂. Choose the correct answer below. OA. Ho: Hd #0 OB. Ho: Hd 20 HA: Hd <0 HA: Hd=0 C. Ho: Hd ≤0 O D. Ho: Hd = 0 HA: Hd #0 HA: Hd>0 Calculate the standardized test statistic. (Round to three decimal places as needed.)
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter10: Statistics
Section10.6: Summarizing Categorical Data
Problem 26PPS
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![### Effect of New Cereal on Total Blood Cholesterol Levels
A food manufacturer claims that incorporating its new cereal into a daily diet can lower total blood cholesterol levels. To verify this claim, data were collected from seven patients before and after one year of consuming the cereal as part of their diets. The table below provides the total blood cholesterol levels (in milligrams per deciliter of blood) for each patient:
| Patient | Total Blood Cholesterol (Before) | Total Blood Cholesterol (After) |
|---------|----------------------------------|---------------------------------|
| 1 | 200 | 196 |
| 2 | 230 | 227 |
| 3 | 240 | 245 |
| 4 | 240 | 236 |
| 5 | 245 | 243 |
| 6 | 260 | 255 |
| 7 | 225 | 223 |
### Hypothesis Testing
To assess whether the new cereal effectively lowers blood cholesterol levels, we define the following:
- Population 1: Blood cholesterol levels before eating the cereal.
- Population 2: Blood cholesterol levels after eating the cereal.
We set up our null and alternative hypotheses as follows:
- Null Hypothesis (H₀): μ_d ≤ 0 (The mean difference in cholesterol levels before and after consuming the cereal is less than or equal to zero, indicating no decrease or an increase in cholesterol levels.)
- Alternative Hypothesis (Hₐ): μ_d > 0 (The mean difference in cholesterol levels before and after consuming the cereal is greater than zero, indicating a decrease in cholesterol levels.)
The correct answer is **C**.
### Calculation of the Standardized Test Statistic
We need to calculate the standardized test statistic (t) to determine whether the observed data provide sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis.
\[ t = \text{(formula here)} \]
**(The actual formula and calculation steps were not provided; hence it's indicated with a placeholder for detailed calculations.)**
**Note:** The t-value should be rounded to three decimal places as needed.
In summary, this analysis involves testing whether the new cereal has a significant effect on lowering total blood cholesterol levels, based on the provided patient data and defined hypotheses.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2efc1b7c-4803-4177-8fdb-49aa376782de%2F979df62a-d92d-4c8a-8e45-bcd52d5ede60%2F5vc86pl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Effect of New Cereal on Total Blood Cholesterol Levels
A food manufacturer claims that incorporating its new cereal into a daily diet can lower total blood cholesterol levels. To verify this claim, data were collected from seven patients before and after one year of consuming the cereal as part of their diets. The table below provides the total blood cholesterol levels (in milligrams per deciliter of blood) for each patient:
| Patient | Total Blood Cholesterol (Before) | Total Blood Cholesterol (After) |
|---------|----------------------------------|---------------------------------|
| 1 | 200 | 196 |
| 2 | 230 | 227 |
| 3 | 240 | 245 |
| 4 | 240 | 236 |
| 5 | 245 | 243 |
| 6 | 260 | 255 |
| 7 | 225 | 223 |
### Hypothesis Testing
To assess whether the new cereal effectively lowers blood cholesterol levels, we define the following:
- Population 1: Blood cholesterol levels before eating the cereal.
- Population 2: Blood cholesterol levels after eating the cereal.
We set up our null and alternative hypotheses as follows:
- Null Hypothesis (H₀): μ_d ≤ 0 (The mean difference in cholesterol levels before and after consuming the cereal is less than or equal to zero, indicating no decrease or an increase in cholesterol levels.)
- Alternative Hypothesis (Hₐ): μ_d > 0 (The mean difference in cholesterol levels before and after consuming the cereal is greater than zero, indicating a decrease in cholesterol levels.)
The correct answer is **C**.
### Calculation of the Standardized Test Statistic
We need to calculate the standardized test statistic (t) to determine whether the observed data provide sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis.
\[ t = \text{(formula here)} \]
**(The actual formula and calculation steps were not provided; hence it's indicated with a placeholder for detailed calculations.)**
**Note:** The t-value should be rounded to three decimal places as needed.
In summary, this analysis involves testing whether the new cereal has a significant effect on lowering total blood cholesterol levels, based on the provided patient data and defined hypotheses.
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