A random sample of the number of unhealthy air days in 10 metropolitan regions, based on the AQI (Air Quality Index), is given below. Construct a 90% confidence interval for the true population mean number of unhealthy air days. Write a sentence to interpret the interval. (Round final values to 2 decimal places). 60 12 9 40 27 38 90 5 13 40

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A random sample of the number of unhealthy air days in 10 metropolitan regions, based on the AQI (Air Quality
Index), is given below. Construct a 90% confidence interval for the true population mean number of unhealthy
air days. Write a sentence to interpret the interval. (Round final values to 2 decimal places).

60 12 9 40 27 38 90 5 13 40

 

This image contains a t-distribution critical values table (Table D) along with a diagram illustrating the t-distribution curve. Here’s a detailed breakdown:

### Diagram Explanation:
- **T-Distribution Curve**: 
  - The diagram shows a bell-shaped t-distribution curve.
  - It highlights an area under the curve representing the upper-tail probability \( p \).
  - The critical value \( t^* \) is marked on the horizontal axis. This value separates the shaded area (representing probability \( p \)) from the rest of the curve.
  - The text explains that the table entry for \( p \) and \( C \) is the critical value \( t^* \) with probability \( p \) lying to its right and probability \( C \) lying between \(-t^*\) and \( t^* \).

### Table D Explanation: t distribution critical values
- **Columns**: 
  - The columns are labeled with different upper-tail probabilities \( p \): .25, .20, .15, .10, .05, .025, .02, .01, .005, .0025, .001, .0005.
  
- **Rows**:
  - The first column lists the degrees of freedom \( df \).
  - The table entries provide the critical values \( t^* \) corresponding to each degree of freedom and upper-tail probability.

- **Bottom Row**:
  - Labeled as \( z^* \), providing the critical values for different confidence levels \( C \) ranging from 50% to 99.9%.

This table and diagram are crucial tools for statistical analysis, helping to determine the critical value \( t^* \) for conducting hypothesis tests and constructing confidence intervals based on the t-distribution.
Transcribed Image Text:This image contains a t-distribution critical values table (Table D) along with a diagram illustrating the t-distribution curve. Here’s a detailed breakdown: ### Diagram Explanation: - **T-Distribution Curve**: - The diagram shows a bell-shaped t-distribution curve. - It highlights an area under the curve representing the upper-tail probability \( p \). - The critical value \( t^* \) is marked on the horizontal axis. This value separates the shaded area (representing probability \( p \)) from the rest of the curve. - The text explains that the table entry for \( p \) and \( C \) is the critical value \( t^* \) with probability \( p \) lying to its right and probability \( C \) lying between \(-t^*\) and \( t^* \). ### Table D Explanation: t distribution critical values - **Columns**: - The columns are labeled with different upper-tail probabilities \( p \): .25, .20, .15, .10, .05, .025, .02, .01, .005, .0025, .001, .0005. - **Rows**: - The first column lists the degrees of freedom \( df \). - The table entries provide the critical values \( t^* \) corresponding to each degree of freedom and upper-tail probability. - **Bottom Row**: - Labeled as \( z^* \), providing the critical values for different confidence levels \( C \) ranging from 50% to 99.9%. This table and diagram are crucial tools for statistical analysis, helping to determine the critical value \( t^* \) for conducting hypothesis tests and constructing confidence intervals based on the t-distribution.
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