---### 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-ValuesIn 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 Representation1. **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 TableThe 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

Given: Sample size n = 50 sample mean = 17.598 sample standard deviation = 16.017127

Answered: a. What is the number of degrees of…

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

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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Question

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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

### 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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