Salt sensitivities for ten patients before and after antihypertensive treatment are below.Patient12345678910Before6.42-6.711.8611.409.391.449.9722.867.7415.49After10.7011.402.0910.196.99-0.773.296.11-4.028.04(a) Are the conditions met for running a paired t-test? Explain, addressing each condition. If you need to check for normality, use R to perform the usual diagnostic checks.(b) Regardless of your answer to part (a), run a paired t-test with R. Be sure to state the null and alternative hypotheses, the test statistic, the p-value, and your conclusion.(c) What is a good estimate for the mean sensitivity change? ### Educational Content: Salt Sensitivity Analysis for Patients#### Data OverviewThe table below shows salt sensitivities for ten patients before and after antihypertensive treatment. Sensitivities are recorded as numerical values, each corresponding to a specific patient.| Patient | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 ||---------|------|-------|------|-------|------|-------|------|------|-------|------|| Before | 6.42 | -6.71 | 1.86 | 11.40 | 9.39 | 1.44 | 9.97 | 22.86| 7.74 | 15.49|| After | 10.70| 11.40 | 2.09 | 10.19 | 6.99 | -0.77 | 3.29 | 6.11 | -4.02 | 8.04 |#### Analysis Questions**(a) Conditions for Paired t-Test**Are the conditions met for running a paired t-test? Consider these aspects:- **Paired Design**: Each measurement before treatment is paired with a measurement after treatment for the same patient.- **Normality**: Check the distribution of the differences (after - before) for normality, often using diagnostic plots or statistical tests.**(b) Running a Paired t-Test with R**In R, perform a paired t-test. Define:- **Null Hypothesis (H₀)**: The antihypertensive treatment has no effect on salt sensitivity (mean difference = 0).- **Alternative Hypothesis (H₁)**: The treatment affects salt sensitivity (mean difference ≠ 0).- Provide the test statistic, p-value, and conclude whether to accept or reject H₀.**(c) Estimation of Mean Sensitivity Change**Calculate and interpret the mean change in sensitivity from before to after the treatment to provide an estimate of the treatment's impact.This exercise helps to understand the use of statistical tests in evaluating medical treatments and their effects on patients.

Solutions : (A) Yes, the conditions met for running a paired t-test as : The sensitivities for ten…

Answered: (a) Are the conditions met for running…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

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Salt sensitivities for ten patients before and after antihypertensive treatment are below.
Patient
1
2
3
4
5
6
7
8
9
10
Before
6.42
-6.71
1.86
11.40
9.39
1.44
9.97
22.86
7.74
15.49
After
10.70
11.40
2.09
10.19
6.99
-0.77
3.29
6.11
-4.02
8.04
(a) Are the conditions met for running a paired t-test? Explain, addressing each condition. If you need to check for normality, use R to perform the usual diagnostic checks.
(b) Regardless of your answer to part (a), run a paired t-test with R. Be sure to state the null and alternative hypotheses, the test statistic, the p-value, and your conclusion.
(c) What is a good estimate for the mean sensitivity change?

### Educational Content: Salt Sensitivity Analysis for Patients

#### Data Overview
The table below shows salt sensitivities for ten patients before and after antihypertensive treatment. Sensitivities are recorded as numerical values, each corresponding to a specific patient.

| Patient | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---------|------|-------|------|-------|------|-------|------|------|-------|------|
| Before | 6.42 | -6.71 | 1.86 | 11.40 | 9.39 | 1.44 | 9.97 | 22.86| 7.74 | 15.49|
| After | 10.70| 11.40 | 2.09 | 10.19 | 6.99 | -0.77 | 3.29 | 6.11 | -4.02 | 8.04 |

#### Analysis Questions

**(a) Conditions for Paired t-Test**

Are the conditions met for running a paired t-test? Consider these aspects:
- **Paired Design**: Each measurement before treatment is paired with a measurement after treatment for the same patient.
- **Normality**: Check the distribution of the differences (after - before) for normality, often using diagnostic plots or statistical tests.

**(b) Running a Paired t-Test with R**

In R, perform a paired t-test. Define:
- **Null Hypothesis (H₀)**: The antihypertensive treatment has no effect on salt sensitivity (mean difference = 0).
- **Alternative Hypothesis (H₁)**: The treatment affects salt sensitivity (mean difference ≠ 0).
- Provide the test statistic, p-value, and conclude whether to accept or reject H₀.

**(c) Estimation of Mean Sensitivity Change**

Calculate and interpret the mean change in sensitivity from before to after the treatment to provide an estimate of the treatment's impact.

This exercise helps to understand the use of statistical tests in evaluating medical treatments and their effects on patients.

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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