The average annual miles driven per vehicle in the United States is 11.1 thousand miles, with σ ≈ 600 miles. Suppose that a random sample of 36 vehicles owned by residents of Chicago showed that the average mileage driven last year was 10.9 thousand miles. Does this indicate that the average miles driven per vehicle in Chicago is different from (higher or lower than) the national average? Use a 0.05 level of significance. What are we testing in this problem? State the null and alternate hypotheses. What sampling distribution will you use? What assumptions are you making?The standard normal, since we assume that x has a normal distribution with known σ.The Student's t, since we assume that x has a normal distribution with unknown σ. The standard normal, since we assume that x has a normal distribution with unknown σ.The Student's t, since we assume that x has a normal distribution with known σ. What is the value of the sample test statistic? (Round your answer to two decimal places.) (c) Find (or estimate) the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α? At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant. At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant.At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant. Interpret your conclusion in the context of the application. There is sufficient evidence at the 0.05 level to conclude that the miles driven per vehicle in the city differs from the national average.There is insufficient evidence at the 0.05 level to conclude that the miles driven per vehicle in the city differs from the national average. ### Understanding Sampling Distributions and P-ValuesIn this exercise, we aim to illustrate how to sketch the sampling distribution and identify the area corresponding to the P-value. Below are different visual representations of sampling distributions with shaded regions indicating various areas.#### Diagram Descriptions:1. **Top-left Diagram**: - **Description**: A bell-shaped normal distribution curve centered at zero, extending between -4 and 4 on the x-axis. - **Shaded Area**: The central part of the distribution is shaded from -2 to 2. This represents a common confidence interval, typically capturing a certain percentage (e.g., 95%) of the data if standard normal distribution applies. - **Interpretation**: This shaded area indicates a scenario where a large proportion of the sampling distribution falls within these bounds, reflecting a high-confidence interval in a hypothesis test.2. **Top-right Diagram**: - **Description**: Similarly, a bell-shaped normal distribution curve centered at zero, extending between -4 and 4 on the x-axis. - **Shaded Area**: There are two narrow shaded areas in the tails of the distribution, one between -2 and -1.5 and the other between 1.5 and 2. This represents the areas in the tails, which are usually associated with low probabilities (the P-values). - **Interpretation**: This represents a two-tailed test case where the P-value is derived from the extreme ends (tails) of the distribution, often implying critical regions in hypothesis testing.3. **Bottom-left Diagram**: - **Description**: Another bell-shaped normal distribution curve centered at zero, extending between -4 and 4 on the x-axis. - **Shaded Area**: The area shaded here is from zero to 4, covering the right half of the distribution. - **Interpretation**: This shading represents a one-tailed test where the P-value is in the right end of the distribution, used when testing hypotheses for deviations in one direction (positive).4. **Bottom-right Diagram**: - **Description**: Again, a bell-shaped normal distribution curve centered at zero, extending between -4 and 4 on the x-axis. - **Shaded Area**: The shaded area here is the left half of the distribution from -4 to 0. - **Interpretation**: This reflects a one-tailed test for the

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Description: The average annual miles driven per vehicle in the United States is 11.1 thousand miles, with σ ≈ 600 miles. Suppose that a random sample of 36 vehicles owned by residents of Chicago showed that the average mileage driven last year was 10.9 thousand

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The average annual miles driven per vehicle in the United States is 11.1 thousand miles, with σ ≈ 600 miles. Suppose that a random sample of 36 vehicles owned by residents of Chicago showed that the average mileage driven last year was 10.9 thousand miles. Does this indicate that the average miles driven per vehicle in Chicago is different from (higher or lower than) the national average? Use a 0.05 level of significance. What are we testing in this problem? State the null and alternate hypotheses. What sampling distribution will you use? What assumptions are you making? The standard normal, since we assume that x has a normal distribution with known σ. The Student's t, since we assume that x has a normal distribution with unknown σ. The standard normal, since we assume that x has a normal distribution with unknown σ. The Student's t, since we assume that x has a normal distribution with known σ. What is the value of the sample test statistic? (Round your answer to two decimal places.) (c) Find (or estimate) the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α? At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant. At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant. At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant. At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant. Interpret your conclusion in the context of the application. There is sufficient evidence at the 0.05 level to conclude that the miles driven per vehicle in the city differs from the national average. There is insufficient evidence at the 0.05 level to conclude that the miles driven per vehicle in the city differs from the national average.

The average annual miles driven per vehicle in the United States is 11.1 thousand miles, with σ ≈ 600 miles. Suppose that a random sample of 36 vehicles owned by residents of Chicago showed that the average mileage driven last year was 10.9 thousand miles. Does this indicate that the average miles driven per vehicle in Chicago is different from (higher or lower than) the national average? Use a 0.05 level of significance. What are we testing in this problem? State the null and alternate hypotheses. What sampling distribution will you use? What assumptions are you making? The standard normal, since we assume that x has a normal distribution with known σ. The Student's t, since we assume that x has a normal distribution with unknown σ. The standard normal, since we assume that x has a normal distribution with unknown σ. The Student's t, since we assume that x has a normal distribution with known σ. What is the value of the sample test statistic? (Round your answer to two decimal places.) (c) Find (or estimate) the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α? At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant. At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant. At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant. At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant. Interpret your conclusion in the context of the application. There is sufficient evidence at the 0.05 level to conclude that the miles driven per vehicle in the city differs from the national average. There is insufficient evidence at the 0.05 level to conclude that the miles driven per vehicle in the city differs from the national average.

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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What are we testing in this problem?

State the null and alternate hypotheses.

What sampling distribution will you use? What assumptions are you making?

The standard normal, since we assume that x has a normal distribution with known σ.

The Student's t, since we assume that x has a normal distribution with unknown σ.

The standard normal, since we assume that x has a normal distribution with unknown σ.

The Student's t, since we assume that x has a normal distribution with known σ.

What is the value of the sample test statistic? (Round your answer to two decimal places.)

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?

At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.

At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant.

At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant.

At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

Interpret your conclusion in the context of the application.

There is sufficient evidence at the 0.05 level to conclude that the miles driven per vehicle in the city differs from the national average.

There is insufficient evidence at the 0.05 level to conclude that the miles driven per vehicle in the city differs from the national average.

### Understanding Sampling Distributions and P-Values

In this exercise, we aim to illustrate how to sketch the sampling distribution and identify the area corresponding to the P-value. Below are different visual representations of sampling distributions with shaded regions indicating various areas.

#### Diagram Descriptions:

1. **Top-left Diagram**:
- **Description**: A bell-shaped normal distribution curve centered at zero, extending between -4 and 4 on the x-axis.
- **Shaded Area**: The central part of the distribution is shaded from -2 to 2. This represents a common confidence interval, typically capturing a certain percentage (e.g., 95%) of the data if standard normal distribution applies.
- **Interpretation**: This shaded area indicates a scenario where a large proportion of the sampling distribution falls within these bounds, reflecting a high-confidence interval in a hypothesis test.

2. **Top-right Diagram**:
- **Description**: Similarly, a bell-shaped normal distribution curve centered at zero, extending between -4 and 4 on the x-axis.
- **Shaded Area**: There are two narrow shaded areas in the tails of the distribution, one between -2 and -1.5 and the other between 1.5 and 2. This represents the areas in the tails, which are usually associated with low probabilities (the P-values).
- **Interpretation**: This represents a two-tailed test case where the P-value is derived from the extreme ends (tails) of the distribution, often implying critical regions in hypothesis testing.

3. **Bottom-left Diagram**:
- **Description**: Another bell-shaped normal distribution curve centered at zero, extending between -4 and 4 on the x-axis.
- **Shaded Area**: The area shaded here is from zero to 4, covering the right half of the distribution.
- **Interpretation**: This shading represents a one-tailed test where the P-value is in the right end of the distribution, used when testing hypotheses for deviations in one direction (positive).

4. **Bottom-right Diagram**:
- **Description**: Again, a bell-shaped normal distribution curve centered at zero, extending between -4 and 4 on the x-axis.
- **Shaded Area**: The shaded area here is the left half of the distribution from -4 to 0.
- **Interpretation**: This reflects a one-tailed test for the

Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.

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