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process for the other sample. t Distribution Degrees of Freedom = 52 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 AA What is the probability of getting the t statistic or something more extreme for the sample size of n = 29? p = . What is the probability of getting the t statistic or something more extreme for the sample size of n = 75? p =. The t distribution is with a smaller n. (Hint: To best see this, click the radio button in the tool with no vertical lines. Slowly move the Degrees of Freedom slider from the smallest value to the largest value, and observe how the shape of the distribution changes.)

process for the other sample. t Distribution Degrees of Freedom = 52 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 AA What is the probability of getting the t statistic or something more extreme for the sample size of n = 29? p = . What is the probability of getting the t statistic or something more extreme for the sample size of n = 75? p =. The t distribution is with a smaller n. (Hint: To best see this, click the radio button in the tool with no vertical lines. Slowly move the Degrees of Freedom slider from the smallest value to the largest value, and observe how the shape of the distribution changes.)

MATLAB: An Introduction with Applications
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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**4. The t statistic, the t distribution, and sample size** The average duration of labor from the first contraction to the birth of the baby in women over 35 who have not previously given birth and who do not use any pharmaceuticals is 16 hours. Suppose you have a sample of 29 women over 35 who have not previously given birth who exercise daily, and who have an average duration of labor of 17.3 hours and a sample variance of 37.2 hours. You want to test the hypothesis that women over 35 who have not previously given birth and who exercise daily have a different duration of labor than all women over 35 who have not previously given birth. Calculate the t statistic. To do this, you first need to calculate the estimated standard error. The estimated standard error is \(s_{\text{M}} = 1.1326\). The t statistic is \(1.15\). Now suppose you have a larger sample size \(n = 75\). Calculate the estimated standard error and the t statistic for this sample with the same sample average and the same standard deviation as above, but with the larger sample size. The new estimated standard error is \(0.7043\). The new t statistic is \(1.85\). Note that the t statistic becomes **larger** as \(n\) becomes larger. Use the Distributions tool to look at the t distributions for different sample sizes. To do this, choose the Degrees of Freedom for the first sample size on the slider, and click the radio button with the single orange line. Move the orange vertical line to the right until the number below the orange line is located on the t statistic. The probability of getting that t statistic or one more extreme will appear in the bubble with the orange type. Now repeat the process for the other sample.
Transcribed Image Text:**4. The t statistic, the t distribution, and sample size** The average duration of labor from the first contraction to the birth of the baby in women over 35 who have not previously given birth and who do not use any pharmaceuticals is 16 hours. Suppose you have a sample of 29 women over 35 who have not previously given birth who exercise daily, and who have an average duration of labor of 17.3 hours and a sample variance of 37.2 hours. You want to test the hypothesis that women over 35 who have not previously given birth and who exercise daily have a different duration of labor than all women over 35 who have not previously given birth. Calculate the t statistic. To do this, you first need to calculate the estimated standard error. The estimated standard error is \(s_{\text{M}} = 1.1326\). The t statistic is \(1.15\). Now suppose you have a larger sample size \(n = 75\). Calculate the estimated standard error and the t statistic for this sample with the same sample average and the same standard deviation as above, but with the larger sample size. The new estimated standard error is \(0.7043\). The new t statistic is \(1.85\). Note that the t statistic becomes **larger** as \(n\) becomes larger. Use the Distributions tool to look at the t distributions for different sample sizes. To do this, choose the Degrees of Freedom for the first sample size on the slider, and click the radio button with the single orange line. Move the orange vertical line to the right until the number below the orange line is located on the t statistic. The probability of getting that t statistic or one more extreme will appear in the bubble with the orange type. Now repeat the process for the other sample.
**Educational Content: T Distribution Exploration** **Graph Description:** The graph displays a t-distribution curve with a smooth, symmetric shape centered at 0, illustrating the concept of degrees of freedom in statistics. This particular graph shows a t-distribution with 52 degrees of freedom, indicated by a horizontal slider. As the degrees of freedom increase, the t-distribution approaches a standard normal distribution. The x-axis is labeled from -3.0 to 3.0, and the y-axis is not labeled, as it typically represents probability density. Two radio buttons are shown beneath the slider: - The first option represents the t-distribution visualization with vertical lines. - The second option displays it without vertical lines. There is also a small icon, depicting what looks like a graph with shaded regions, perhaps to indicate critical regions or areas of interest. **Text and Instructions:** "What is the probability of getting the t statistic or something more extreme for the sample size of n = 29? p = ________ ▼. What is the probability of getting the t statistic or something more extreme for the sample size of n = 75? p = ________ ▼. The t distribution is _______________ ▼ with a smaller n. (Hint: To best see this, click the radio button in the tool with no vertical lines. Slowly move the Degrees of Freedom slider from the smallest value to the largest value, and observe how the shape of the distribution changes.)" **Interactive Component:** Students can interact with the graph by adjusting the degrees of freedom using the slider and selecting different visualization options with the radio buttons. This helps them observe how changing degrees of freedom affect the shape of the t-distribution and better understand the relationship between sample size and probability in statistical analysis.
Transcribed Image Text:**Educational Content: T Distribution Exploration** **Graph Description:** The graph displays a t-distribution curve with a smooth, symmetric shape centered at 0, illustrating the concept of degrees of freedom in statistics. This particular graph shows a t-distribution with 52 degrees of freedom, indicated by a horizontal slider. As the degrees of freedom increase, the t-distribution approaches a standard normal distribution. The x-axis is labeled from -3.0 to 3.0, and the y-axis is not labeled, as it typically represents probability density. Two radio buttons are shown beneath the slider: - The first option represents the t-distribution visualization with vertical lines. - The second option displays it without vertical lines. There is also a small icon, depicting what looks like a graph with shaded regions, perhaps to indicate critical regions or areas of interest. **Text and Instructions:** "What is the probability of getting the t statistic or something more extreme for the sample size of n = 29? p = ________ ▼. What is the probability of getting the t statistic or something more extreme for the sample size of n = 75? p = ________ ▼. The t distribution is _______________ ▼ with a smaller n. (Hint: To best see this, click the radio button in the tool with no vertical lines. Slowly move the Degrees of Freedom slider from the smallest value to the largest value, and observe how the shape of the distribution changes.)" **Interactive Component:** Students can interact with the graph by adjusting the degrees of freedom using the slider and selecting different visualization options with the radio buttons. This helps them observe how changing degrees of freedom affect the shape of the t-distribution and better understand the relationship between sample size and probability in statistical analysis.
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