From generation to generation, the mean age when smokers first start to smoke varies. However, the standard deviation of that age remains constant at around 2.1 years. A survey of 42 smokers of this generation was done to see if the mean starting age is at least 19. The sample mean was 18.2 with a sample standard deviation of 1.3. Do the data support the claim at the 5% level?Note: If you are using a Student's t-distribution for the problem, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.)I need help solving the problems in the pictures. ### Hypothesis Testing and Confidence Intervals#### Instructions for Hypothesis Testing:1. **Indicate the correct decision ("reject" or "do not reject" the null hypothesis), the reason for it, and write an appropriate conclusion.** - **(i) Alpha (α):** - Enter an exact number as an integer, fraction, or decimal. - **(ii) Decision:** - ☐ Reject the null hypothesis - ☐ Do not reject the null hypothesis - **(iii) Reason for Decision:** - ☐ Since α < p-value, we reject the null hypothesis. - ☐ Since α > p-value, we do not reject the null hypothesis. - ☐ Since α < p-value, we do not reject the null hypothesis. - ☐ Since α > p-value, we reject the null hypothesis. - **(iv) Conclusion:** - ☐ There is sufficient evidence to conclude that the starting age for smoking in this generation is less than 19. - ☐ There is not sufficient evidence to conclude that the starting age for smoking in this generation is less than 19.#### Part (i): Confidence Interval Construction- **Objective:** Construct a 95% confidence interval for the true mean.- **Steps:** - Sketch the graph of the situation. - Label the point estimate and the lower and upper bounds of the confidence interval. - Round your lower and upper bounds to two decimal places.- **Graph Explanation:** The graph is a normal distribution curve depicting a 95% confidence interval (95% C.I.). The confidence interval is symmetrical around the central mean value: - **Point Estimate:** Represents the central value of the interval. - **Lower Bound:** 17.6 - **Upper Bound:** 18.8 - **Width of Interval:** 1.2 The curve visually demonstrates the range within which the true mean is expected to fall, with 95% certainty. ### Hypothesis Testing: One Sample#### Part (d)State the distribution to use for the test. (Round your answers to four decimal places.)\[ \bar{X} \sim z \left(\text{______, ______}\right) \]#### Part (e)What is the test statistic? (If using the z distribution, round your answers to two decimal places, and if using the t distribution, round your answers to three decimal places.)\[ z = -2.4096 \]#### Part (f)What is the p-value? (Round your answer to four decimal places.)\[ \text{p-value:} \; \_ \]**Explain what the p-value means for this problem:**- ( ) If $ H_0 $ is false, then there is a chance equal to the p-value that the average age of people when they first begin to smoke is not 18.2 years or less.- ( ) If $ H_0 $ is true, then there is a chance equal to the p-value that the average age of people when they first begin to smoke is not 18.2 years or less.- ( ) If $ H_0 $ is false, then there is a chance equal to the p-value that the average age of people when they first begin to smoke is 18.2 years or less.- ( ) If $ H_0 $ is true, then there is a chance equal to the p-value that the average age of people when they first begin to smoke is 18.2 years or less.#### Part (g)Sketch a picture of this situation. Label and scale the horizontal axis and shade the region(s) corresponding to the p-value.**Diagrams:**1. **First Distribution Diagram:** - **Horizontal Axis:** Labeled with $ \mu $ and $\bar{X}$. - **Normal Curve:** Symmetrical. - **Shaded Region:** Indicates the p-value on the left side.2. **Second Distribution Diagram (Highlighted):** - **Horizontal Axis:** Labeled with $ \mu $ and $\bar{X}$. - **Normal Curve:** Symmetrical. - **Shaded Region:** Indicates the p-value on the right side.3. **Third Distribution Diagram:** - **Horizontal Axis:** Labeled with $ \mu $ and \(\

Given Information: Standard deviation σ = 2.1 years. Sample size (n) = 42 Sample mean (x¯) = 18.2…

Answered: From generation to generation, the mean…

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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From generation to generation, the mean age when smokers first start to smoke varies. However, the standard deviation of that age remains constant at around 2.1 years. A survey of 42 smokers of this generation was done to see if the mean starting age is at least 19. The sample mean was 18.2 with a sample standard deviation of 1.3. Do the data support the claim at the 5% level?

Note: If you are using a Student's t-distribution for the problem, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.)

I need help solving the problems in the pictures.

### Hypothesis Testing and Confidence Intervals

#### Instructions for Hypothesis Testing:

1. **Indicate the correct decision ("reject" or "do not reject" the null hypothesis), the reason for it, and write an appropriate conclusion.**

- **(i) Alpha (α):**
- Enter an exact number as an integer, fraction, or decimal.

- **(ii) Decision:**
- ☐ Reject the null hypothesis
- ☐ Do not reject the null hypothesis

- **(iii) Reason for Decision:**
- ☐ Since α < p-value, we reject the null hypothesis.
- ☐ Since α > p-value, we do not reject the null hypothesis.
- ☐ Since α < p-value, we do not reject the null hypothesis.
- ☐ Since α > p-value, we reject the null hypothesis.

- **(iv) Conclusion:**
- ☐ There is sufficient evidence to conclude that the starting age for smoking in this generation is less than 19.
- ☐ There is not sufficient evidence to conclude that the starting age for smoking in this generation is less than 19.

#### Part (i): Confidence Interval Construction

- **Objective:** Construct a 95% confidence interval for the true mean.

- **Steps:**
- Sketch the graph of the situation.
- Label the point estimate and the lower and upper bounds of the confidence interval.
- Round your lower and upper bounds to two decimal places.

- **Graph Explanation:**

The graph is a normal distribution curve depicting a 95% confidence interval (95% C.I.). The confidence interval is symmetrical around the central mean value:

- **Point Estimate:** Represents the central value of the interval.
- **Lower Bound:** 17.6
- **Upper Bound:** 18.8
- **Width of Interval:** 1.2

The curve visually demonstrates the range within which the true mean is expected to fall, with 95% certainty.

### Hypothesis Testing: One Sample

#### Part (d)
State the distribution to use for the test. (Round your answers to four decimal places.)

\[ \bar{X} \sim z \left(\text{______, ______}\right) \]

#### Part (e)
What is the test statistic? (If using the z distribution, round your answers to two decimal places, and if using the t distribution, round your answers to three decimal places.)

\[ z = -2.4096 \]

#### Part (f)
What is the p-value? (Round your answer to four decimal places.)

\[ \text{p-value:} \; \_ \]

**Explain what the p-value means for this problem:**

- ( ) If $ H_0 $ is false, then there is a chance equal to the p-value that the average age of people when they first begin to smoke is not 18.2 years or less.
- ( ) If $ H_0 $ is true, then there is a chance equal to the p-value that the average age of people when they first begin to smoke is not 18.2 years or less.
- ( ) If $ H_0 $ is false, then there is a chance equal to the p-value that the average age of people when they first begin to smoke is 18.2 years or less.
- ( ) If $ H_0 $ is true, then there is a chance equal to the p-value that the average age of people when they first begin to smoke is 18.2 years or less.

#### Part (g)
Sketch a picture of this situation. Label and scale the horizontal axis and shade the region(s) corresponding to the p-value.

**Diagrams:**

1. **First Distribution Diagram:**
- **Horizontal Axis:** Labeled with $ \mu $ and $\bar{X}$.
- **Normal Curve:** Symmetrical.
- **Shaded Region:** Indicates the p-value on the left side.

2. **Second Distribution Diagram (Highlighted):**
- **Horizontal Axis:** Labeled with $ \mu $ and $\bar{X}$.
- **Normal Curve:** Symmetrical.
- **Shaded Region:** Indicates the p-value on the right side.

3. **Third Distribution Diagram:**
- **Horizontal Axis:** Labeled with $ \mu $ and \(\

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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