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a. What is the number of degrees of freedom that should be used for finding the critical value tu/2? df =O (Type a whole number.) %3D

a. What is the number of degrees of freedom that should be used for finding the critical value tu/2? df =O (Type a whole number.) %3D

Trigonometry (MindTap Course List)
Trigonometry (MindTap Course List)
8th Edition
ISBN:9781305652224
Author:Charles P. McKeague, Mark D. Turner
Publisher:Charles P. McKeague, Mark D. Turner
Chapter4: Graphing And Inverse Functions
Section: Chapter Questions
Problem 6GP: If your graphing calculator is capable of computing a least-squares sinusoidal regression model, use...
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--- ### Understanding Critical t Values **Refer to the accompanying data display that results from a sample of airport data speeds in Mbps. Complete parts (a) through (c) below.** #### Step-by-Step Instructions ##### a. Determining the Degrees of Freedom - **Question**: What is the number of degrees of freedom that should be used for finding the critical value \( t_{\alpha/2} \)? - **Answer**: - **Given Information for Calculation**: - Confidence Interval (\(TInterval\)): (13.046, 22.15) - Sample Mean (\(x\)): 17.598 - Sample Standard Deviation (\(S_x\)): 16.01712719 - Sample Size (\(n\)): 50 - **Calculation of Degrees of Freedom**: \[ \text{df} = n - 1 = 50 - 1 = 49 \] - **Input Field**: - df = (Type a whole number.) ##### Critical t Values Explanation - **Figure**: Critical t values (right-side image) - **Graph Description**: - The figure displays two bell-shaped curves titled "Left tail" and "Right tail". - Each curve represents the distribution of t-values. - The region under the tail is shaded, indicating the critical area (\(\alpha\)). - Below each curve, there is a line marking the "Critical t value" for the tail. - **Table Description**: - The table on the right side of the figure is a "t Distribution: Critical t Values" table. - The first and last columns show the "Degrees of Freedom". - Various critical t values are listed for different levels of significance (0.01, 0.025, 0.05, 0.10) in both one-tail and two-tail scenarios. ##### How to Use the Table: - Find the row corresponding to the degrees of freedom (e.g., df = 49). - Identify the critical t value by locating the appropriate column for the significance level (\(\alpha\)) of interest. > **Note**: Critical t values are essential for hypothesis testing and constructing confidence intervals. The degrees of freedom depend on the sample size, and accurate identification of df is crucial for correct analysis. --- This transcription is designed to be informative for students
Transcribed Image Text:--- ### Understanding Critical t Values **Refer to the accompanying data display that results from a sample of airport data speeds in Mbps. Complete parts (a) through (c) below.** #### Step-by-Step Instructions ##### a. Determining the Degrees of Freedom - **Question**: What is the number of degrees of freedom that should be used for finding the critical value \( t_{\alpha/2} \)? - **Answer**: - **Given Information for Calculation**: - Confidence Interval (\(TInterval\)): (13.046, 22.15) - Sample Mean (\(x\)): 17.598 - Sample Standard Deviation (\(S_x\)): 16.01712719 - Sample Size (\(n\)): 50 - **Calculation of Degrees of Freedom**: \[ \text{df} = n - 1 = 50 - 1 = 49 \] - **Input Field**: - df = (Type a whole number.) ##### Critical t Values Explanation - **Figure**: Critical t values (right-side image) - **Graph Description**: - The figure displays two bell-shaped curves titled "Left tail" and "Right tail". - Each curve represents the distribution of t-values. - The region under the tail is shaded, indicating the critical area (\(\alpha\)). - Below each curve, there is a line marking the "Critical t value" for the tail. - **Table Description**: - The table on the right side of the figure is a "t Distribution: Critical t Values" table. - The first and last columns show the "Degrees of Freedom". - Various critical t values are listed for different levels of significance (0.01, 0.025, 0.05, 0.10) in both one-tail and two-tail scenarios. ##### How to Use the Table: - Find the row corresponding to the degrees of freedom (e.g., df = 49). - Identify the critical t value by locating the appropriate column for the significance level (\(\alpha\)) of interest. > **Note**: Critical t values are essential for hypothesis testing and constructing confidence intervals. The degrees of freedom depend on the sample size, and accurate identification of df is crucial for correct analysis. --- This transcription is designed to be informative for students
### Understanding Critical t-Values In statistical hypothesis testing, critical t-values are significant for determining whether to reject the null hypothesis. The t-distribution, similar to the normal distribution but with heavier tails, is used when the sample size is small, and the population standard deviation is unknown. Below, we'll guide you through interpreting t-critical values based on the degrees of freedom (df) and the level of significance (\(\alpha\)). #### Graphical Representation 1. **Single Tail Test (Right Tail)** - The first graph represents the critical t-value for a right-tailed test. - The area under the curve to the right of the marked t-value represents the significance level \(\alpha\). - This is used for one-tailed tests where the hypothesis predicts a direction (e.g., greater than a certain value). 2. **Two-Tailed Test** - The second graph depicts a two-tailed test. - The critical region is split between two tails, each with an area of \(\alpha/2\). - This test is used when the hypothesis does not predict a direction, only that the values are different. #### Critical t-Values Table The table below provides critical t-values corresponding to different degrees of freedom (df) and levels of significance (\(\alpha\)). Find the appropriate value by locating the df in the leftmost and rightmost columns. The intersections within the table show the critical t-values for \(\alpha\) = 0.01, 0.02, 0.05, and 0.10. | df | 0.01 | 0.02 | 0.05 | 0.10 | df | |------|-------|-------|-------|-------|------| | 18 | 2.878 | 2.552 | 2.101 | 1.734 | 18 | | 19 | 2.861 | 2.539 | 2.093 | 1.729 | 19 | | 20 | 2.845 | 2.528 | 2.086 | 1.725 | 20 | | 21 | 2.831 | 2.518 | 2.080 | 1.721 | 21 | | 22 | 2.819 | 2.508 | 2
Transcribed Image Text:### Understanding Critical t-Values In statistical hypothesis testing, critical t-values are significant for determining whether to reject the null hypothesis. The t-distribution, similar to the normal distribution but with heavier tails, is used when the sample size is small, and the population standard deviation is unknown. Below, we'll guide you through interpreting t-critical values based on the degrees of freedom (df) and the level of significance (\(\alpha\)). #### Graphical Representation 1. **Single Tail Test (Right Tail)** - The first graph represents the critical t-value for a right-tailed test. - The area under the curve to the right of the marked t-value represents the significance level \(\alpha\). - This is used for one-tailed tests where the hypothesis predicts a direction (e.g., greater than a certain value). 2. **Two-Tailed Test** - The second graph depicts a two-tailed test. - The critical region is split between two tails, each with an area of \(\alpha/2\). - This test is used when the hypothesis does not predict a direction, only that the values are different. #### Critical t-Values Table The table below provides critical t-values corresponding to different degrees of freedom (df) and levels of significance (\(\alpha\)). Find the appropriate value by locating the df in the leftmost and rightmost columns. The intersections within the table show the critical t-values for \(\alpha\) = 0.01, 0.02, 0.05, and 0.10. | df | 0.01 | 0.02 | 0.05 | 0.10 | df | |------|-------|-------|-------|-------|------| | 18 | 2.878 | 2.552 | 2.101 | 1.734 | 18 | | 19 | 2.861 | 2.539 | 2.093 | 1.729 | 19 | | 20 | 2.845 | 2.528 | 2.086 | 1.725 | 20 | | 21 | 2.831 | 2.518 | 2.080 | 1.721 | 21 | | 22 | 2.819 | 2.508 | 2
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