### Testing an Hypothesis about Heights of Sons and Fathers:To test the belief that sons are taller than their fathers, a student randomly selects 13 fathers who have adult male children. She records the height of both the father and son in inches and obtains the following data. Are sons taller than their fathers? Use the α = 0.10 level of significance. Note: A normal probability plot and boxplot of the data indicate that the differences are approximately normally distributed with no outliers.- [Click here to view the table of data.](#)- [Click here to view the table of critical t-values.](#)#### Conditions for the Test:Which conditions must be met by the sample for this test? Select all that apply.- [ ] A. The differences are normally distributed or the sample size is large.- [ ] B. The sampling method results in an independent sample.- [ ] C. The sample size is no more than 5% of the population size.- [ ] D. The sampling method results in a dependent sample.- [ ] E. The sample size must be large.#### Hypotheses for the Test:Let $ d_i = X_i - Y_i $. Write the hypotheses for the test:- $ H_0 $: [Dropdown for Null Hypothesis]- $ H_1 $: [Dropdown for Alternate Hypothesis]#### Calculate the Test Statistic:\[ t_0 = [Input Box] \] (Round to two decimal places as needed.)#### Identify the Critical Value(s) for this Test:Select the correct choice below and fill in all answer boxes within your choice. (Round to two decimal places as needed.)- [ ] A. $-t_{\alpha} = [Input Box]$- [ ] B. $ t_{\alpha} = [Input Box]$- [ ] C. $ t_{\alpha/2} = [Input Box] $ or $ -t_{\alpha/2} = [Input Box]$#### Decision Rule:Compare the critical value to the test statistic. For a two-tailed test, reject the null hypothesis if $ t_0 < -t_{\alpha/2} $ or $ t_0 > t_{\alpha/2} $. For a left-tailed test, reject the null hypothesis if $ t_0 < -t_{\alpha} $. For a right-tailed test, reject the null hypothesis **Table of Critical Values for Student's t-Distribution**The table below provides critical values for the Student's t-distribution with various areas in the right tail. The degrees of freedom range from 1 to 30. The top row identifies the proportion of the distribution in the right tail for each column. A diagram beside the table provides a visual representation, highlighting the area in the right tail of the t-distribution curve.``` t-Distribution Area in Right TailDegrees of .25 .20 .15 0.10 0.05 0.025 0.02 0.01 0.005 0.0025 0.001 0.0005Freedom 1 1.000 1.376 1.963 3.078 6.314 12.706 15.894 31.821 63.657 127.321 318.309 636.619 2 0.816 1.061 1.386 1.886 2.920 4.303 4.849 6.965 9.925 14.089 22.327 31.599 3 0.765 0.978 1.250 1.638 2.353 3.182 3.482 4.541 5.841 7.453 10.215 12.924 4 0.741 0.941 1.190 1.533 2.132 2.776 2.999 3.747 4.604 5.598 7.173 8.610 5 0.727 0.920 1.156 1.476 2.015 2.571 2.757 3.365 4.032 4.773 5.893 6.869 6 0.718 0.906 1.134 1.440 1.943 2.447 2.612 3.143 3.707

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Description: ### Testing an Hypothesis about Heights of Sons and Fathers:To test the belief that sons are taller than their fathers, a student randomly selects 13 fathers who have adult male children. She records the height of both the father and son in inches a

A paired t-test is used when we are interested in the difference between two variables for the same…

Math

Statistics

-Distribution Area in Richt Tail 1 Table of height data Degrees o Freedom 0,25 0,20 0.15 0.10 O816 1061 0.765 0.978 1250 0.941 1.190 Height of Father, X Height of O Son, Y 73.7 71. 69 0.741 68.6 67.6 67.2 68.6 716 0.706 0.889 0.700 0870 70.4 72.8 68.5 67.8 68.3 712 67.7 68.2 70.7 66.4 67.6 69,4 70,7 70 68.1 67 2 '0.8 72 8 0.684 0.856 1.058 1.315 3.067 3.435 33 三= 司基斗

-Distribution Area in Richt Tail 1 Table of height data Degrees o Freedom 0,25 0,20 0.15 0.10 O816 1061 0.765 0.978 1250 0.941 1.190 Height of Father, X Height of O Son, Y 73.7 71. 69 0.741 68.6 67.6 67.2 68.6 716 0.706 0.889 0.700 0870 70.4 72.8 68.5 67.8 68.3 712 67.7 68.2 70.7 66.4 67.6 69,4 70,7 70 68.1 67 2 '0.8 72 8 0.684 0.856 1.058 1.315 3.067 3.435 33 三= 司基斗

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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Concept explainers

Inverse Normal Distribution

The method used for finding the corresponding z-critical value in a normal distribution using the known probability is said to be an inverse normal distribution. The inverse normal distribution is a continuous probability distribution with a family of two parameters.

Mean, Median, Mode

It is a descriptive summary of a data set. It can be defined by using some of the measures. The central tendencies do not provide information regarding individual data from the dataset. However, they give a summary of the data set. The central tendency or measure of central tendency is a central or typical value for a probability distribution.

Z-Scores

A z-score is a unit of measurement used in statistics to describe the position of a raw score in terms of its distance from the mean, measured with reference to standard deviation from the mean. Z-scores are useful in statistics because they allow comparison between two scores that belong to different normal distributions.

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Question

### Testing an Hypothesis about Heights of Sons and Fathers:
To test the belief that sons are taller than their fathers, a student randomly selects 13 fathers who have adult male children. She records the height of both the father and son in inches and obtains the following data. Are sons taller than their fathers? Use the α = 0.10 level of significance. Note: A normal probability plot and boxplot of the data indicate that the differences are approximately normally distributed with no outliers.

- [Click here to view the table of data.](#)
- [Click here to view the table of critical t-values.](#)

#### Conditions for the Test:
Which conditions must be met by the sample for this test? Select all that apply.

- [ ] A. The differences are normally distributed or the sample size is large.
- [ ] B. The sampling method results in an independent sample.
- [ ] C. The sample size is no more than 5% of the population size.
- [ ] D. The sampling method results in a dependent sample.
- [ ] E. The sample size must be large.

#### Hypotheses for the Test:
Let $ d_i = X_i - Y_i $. Write the hypotheses for the test:
- $ H_0 $: [Dropdown for Null Hypothesis]
- $ H_1 $: [Dropdown for Alternate Hypothesis]

#### Calculate the Test Statistic:
\[ t_0 = [Input Box] \] (Round to two decimal places as needed.)

#### Identify the Critical Value(s) for this Test:
Select the correct choice below and fill in all answer boxes within your choice. (Round to two decimal places as needed.)

- [ ] A. $-t_{\alpha} = [Input Box]$
- [ ] B. $ t_{\alpha} = [Input Box]$
- [ ] C. $ t_{\alpha/2} = [Input Box] $ or $ -t_{\alpha/2} = [Input Box]$

#### Decision Rule:
Compare the critical value to the test statistic. For a two-tailed test, reject the null hypothesis if $ t_0 < -t_{\alpha/2} $ or $ t_0 > t_{\alpha/2} $. For a left-tailed test, reject the null hypothesis if $ t_0 < -t_{\alpha} $. For a right-tailed test, reject the null hypothesis

**Table of Critical Values for Student's t-Distribution**

The table below provides critical values for the Student's t-distribution with various areas in the right tail. The degrees of freedom range from 1 to 30. The top row identifies the proportion of the distribution in the right tail for each column. A diagram beside the table provides a visual representation, highlighting the area in the right tail of the t-distribution curve.

```
t-Distribution
Area in Right Tail
Degrees of .25 .20 .15 0.10 0.05 0.025 0.02 0.01 0.005 0.0025 0.001 0.0005
Freedom
1 1.000 1.376 1.963 3.078 6.314 12.706 15.894 31.821 63.657 127.321 318.309 636.619
2 0.816 1.061 1.386 1.886 2.920 4.303 4.849 6.965 9.925 14.089 22.327 31.599
3 0.765 0.978 1.250 1.638 2.353 3.182 3.482 4.541 5.841 7.453 10.215 12.924
4 0.741 0.941 1.190 1.533 2.132 2.776 2.999 3.747 4.604 5.598 7.173 8.610
5 0.727 0.920 1.156 1.476 2.015 2.571 2.757 3.365 4.032 4.773 5.893 6.869
6 0.718 0.906 1.134 1.440 1.943 2.447 2.612 3.143 3.707

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