Question

Can someone please explain these problems, I don't understand, please and thank you!!

The patients in the Digoxin trial dataset can be considered a population of people suffering from heart failure. These patients were examined before the drug trial began, and their heart rate and blood pressure were recorded. The mean and standard deviation of the variables are listed below. Each variable follows a normal distribution.

Heart rate (beats/min)                          μ = 78.8            σ = 12.66

Systolic blood pressure (mmHg)             μ = 125.8          σ = 19.94

1. A patient with a heart rate of 90 is

1. in the middle 68% of values
2. in the middle 95% of values (but not the middle 68%)
3. in the 5% most extreme values

2. The 10% of patients with the highest heart rates have z scores greater than 1.28 What heart rate does this value correspond to? Report your answer to 2 decimal places.

3. Many random samples of 25 patients are selected from this population, and the mean of their systolic blood pressures is calculated. The many sample means make up a sampling distribution. What is the mean of the sampling distribution (its center)? In other words, what is the mean of the many sample means? Report your answer to 2 decimal places.

4. If μ is the mean from the previous question, 68% of the sample means have values between

1. μ – 0.798 and μ + 0.798
2. μ – 3.988 and μ + 3.988
3. μ – 7.976 and μ + 7.976
4. μ – 19.94 and μ + 19.94

5. The 10% of samples with the highest mean systolic blood pressure have z scores greater than 1.282. What mean blood pressure does this correspond to?

Answer 1

Mean heart rate = 90

z = (Mean - μ)/σ

z = (90-78.8)/12.66

z = 0.88

According to empirical rule, 68% of data falls within the first standard deviation from the mean. In this case, z represents distance of 90 from mean that is 0.88. Since this is less than 1, it shows that a patient with a heart rate of 90 is the middle 68% of values (Option A).

Let x be the heart rate value corresponding to z = 1.28

z = (x-μ)/σ

1.28 = (x-78.8)/12.66

x = 78.8+1.28*12.66

x = 95.00

This value correspond to heart rate of 95.00

Mean of the sampling distribution of sample means = Population mean of Systolic blood pressure = μ_x̄ = 125.8

Standard deviation of the sampling distribution of sample means = σ_x̄ = σ/sqrt(n) = 19.94/sqrt(25) = 3.988

According to empirical rule, 68% of data falls within the first standard deviation from the mean.

Thus, 68% of the sample means have values between μ - σ_x̄ and μ + σ_x̄

Thus, 68% of the sample means have values between μ - 3.988and μ + 3.988 (Option B)

Let x be the mean blood pressure value corresponding to z = 1.28

z = (x-μ)/σ

1.282 = (x-125.8)/19.94

x = 125.8+1.282*19.94

x = 151.363

This value correspond to the mean blood pressure of 151.36