Problems

Scenario 1: Cholesterol

Cholesterolis a type of fat found inthe blood. It is measured as a concentration: the number of milligrams of cholesterol found per deciliter of blood (mg/dL). A high level of total cholesterol in the bloodstream increasesriskforheart disease. For this problem, assume cholesterol in men and women follows a normal distribution, and that "adult man" and "adult woman" refers to a man/woman in the U.S. over age 20. For adult men, total cholesterol has ameanof 188 mg/dL and a standard deviation of 43 mg/dL. For adult women, total cholesterol has ameanof 193 mg/dL and a standard deviation of 42 mg/dL. The CDC defines "high cholesterol" as having total cholesterol of 240 mg/dL or higher, "borderline high" as having a total cholesterol of more than 200 but less than 240, and "healthy" as having total cholesterol of 200 or less. A study published in 2017 indicated that about 11.3% of adult men and 13.2% of adult women have high cholesterol. Use this scenario for problems 1 - 8.

1. We will consider several random variables for this scenario in the following problems. For each random variable below, determine if it follows a normal distribution, a binomial distribution, or neither. (2ptea)

A. Normal B. Binomial C. Neither ___ The total cholesterol in U.S. men over age 20.

___ The sampling distribution of sample means for when 25 random adult women aresampledand their total cholesterol is measured.

___ In a group of 25 randomly chosen adult men, the number who have high cholesterol.

___ For adult women, the total cholesterol in mg/dL.

___ The total cholesterol is measured for each person in a random group of 20 adults (both men and women). Then, the samplemeanis calculated for that group. The sampling distribution of sample means for the groups if this process were repeated.

___ The number of adult women from a random group of 36 who do not have high cholesterol.

___ If a group of 50 random adults (both men and women) is selected, their total cholesterol is measured, and the sample mean is calculated, the sampling distribution of sample means for groups of this size.

2. An adult woman is randomlyselectedand her total cholesterol is measured. What is the probability it is in the "borderline high" range? (4pt)

3. In a study of 45 randomly selected adult men, the number who have high cholesterol is counted. (Assume that 11.3% of men have high cholesterol.)

a. How many of these 45 men do you expect to have high cholesterol? (Round to 1 decimal place.) (3pt) b. What is the standard deviation for the number of these 45 men that have high cholesterol? (Round to 1 decimal place.) (3pt)

c. What is the probability that at least 10 of these 45 men will have high cholesterol? (3pt)

4. The CDC guidelines for cholesterol health are applied to both men and women, but men and women have different distributions of total cholesterol. a. What approximate percent of women have a total cholesterol that would be considered "healthy?" (For this problem, give your answer as a percent, not a decimal. Round your answer to one decimal place.) (3pt)

b. What approximate percent of men have a total cholesterol that would be considered "healthy?" (For this problem, give your answer as a percent, not a decimal. Round your answer to one decimal place.) (3pt)

5. A group of 256 randomly chosen adult menisselected. How many of them do you expect to have a total cholesterol of less than 200 mg/dL? (Round your answer to one decimal place.) (3pt)

6. Oatmeal is a food that is high in fiber and low in fat. A dietician says, "People who regularly eat oatmeal tend to have lower cholesterol." A group of 121 randomly selected adult women who regularly eat oatmeal has a sample mean total cholesterol of 185 mg/dL.

a. What is the probability a randomly selected group of adult women has a sample mean total cholesterol of 185 or less? (4pt)

b. Would this be a significant result? (Chooseone.)(4pt) A. Yes B. No C. Not enough information

7. A researcher measures the total cholesterol of a randomly selected group of 36 adultwomen, andcounts the number of them who have high cholesterol. (Assume that 13.2% of adult women have high cholesterol.)

a. What is the probability that exactly 4 of these 36 women have high cholesterol? (3pt)

b. What is the probability that 8 or less of these 36 women have high cholesterol? (3pt)

8. A doctor recommends drastic lifestyle changes for all adults who are in the top 5% of total cholesterol levels.

a. What total cholesterol level is the cutoff for the top 5% of women? (Round to 1 decimal place.) (4pt)

b. What total cholesterol level is the cutoff for the top 5% of men? (Round to 1 decimal place.) (4pt)

Scenario 2: Electronic Component

The manufacturer of a certain electronic component claims that they are designed to last just slightly more than 4 years because they believe that customers typically replace their device before then. Based on information provided by the company, the components should last a mean of 4.24 years with a standard deviation of 0.45 years. For this scenario, assume the lifespans of this component follow a normal distribution. Use this scenario for problems 9 - 13.

9. We will consider several random variables for this scenario in the following problems. For each random variable below, determine if it follows a normal distribution, a binomial distribution, or neither. (2ptea) A.NormalB. Binomial C. Neither

___ The lifespan of one randomly chosen component.

___ In a group of 35 randomly chosen components, the number of them which last more than 4 years. ___ The sample mean lifespan for groups of 20 randomly chosen components.

___ The sample mean lifespan for groups of 35 randomly chosen components.

___ The number of components that last more than 4.5 years from a group of 10 independent components.

10. The company offers a warranty for this component that allows customers toreturnfor arefund itif it fails in less than 4 years. What is the probability that a randomly chosen component will last less than 4 years? (4pt)

11. The company considers a component to be "successful" if it lasts longer than the warranty period before failing. They estimate that about 70.3% of components last more than 4 years. They find a random group of 10 components that were sold and count the number of them which were "successful," lasting more than 4 years.

a. What is the expected number of these 10 components that will last more than 4 years? (Round your answer to 1 decimal place.) (3pt)

b. What is the standard deviation for the number of these 10 components that will last more than 4 years? (Round your answer to 1 decimal place.) (3pt)

c. What is the probability that at least 8 of these components will last more than 4 years? (3pt)

d. What is the probability that no more than 6 of these components will last more than 4 years? (3pt)

12. The company is not satisfied with how many returns they are processing, and accountants in the company arerecommendingthat they change the warranty period. The accountants suggest basing the period of the warranty on making sure, in the long run, only about 5% of customers will return the component. What amount of time corresponds to the shortest 5% of lifespans for this component? (Round your answer to 1 decimal place.) (4pt)

13. A journalist believes that, in reality, thesecomponents have a shorter lifespan than what the company is reporting. He tests a random group of 35 of these components and finds a mean lifespan of 4.05 years.

a. What is themeanof the sampling distribution of sample means when 35 random components are tested? (2pt)

b. What is the standard error of the sampling distribution when 35 random components are tested? (2pt)

c. If the company's reported lifespan distribution werereally true, what is the probability that a random group of 35 will have a sample mean lifespan of 4.05 years or less? (4pt) d. Would this be a significant result? (Chooseone.)(4pt) A. Yes B. No C. Not enough information