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?

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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.)

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### 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.
Transcribed Image Text:### 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 \(\
Transcribed Image Text:### 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 \(\
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