wo independent simple random samples of professional NBA and NFL players heights are given below.



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Let by the population mean height for NBA players and be the population mean for NFL players. Find a 92% confidence interval . (Round answers to two decimal places.)

margin of error:
lower limit:
upper limit:


Examine the confidence interval and explain what it means in the context of this problem. At the 92% level of confidence, do NBA players tend to have a higher population mean height than NFL players?

    Yes, since the interval contains only positive numbers, we can say that NBA players have a higher population mean height than NFL players.
    No, since the interval contains only negative numbers, we can say that NBA players have a lower population mean height than NFL players.
    It's not clear, since the interval contains both negative and positive numbers, we can't say that NBA players have a higher population mean height than NFL players.



To solve this problem, let's break down the steps required to find the 92% confidence interval for the difference in mean heights between NBA and NFL players.

1. **Calculate the Difference in Sample Means**:
   Let:
   - \(\bar{x}_1\) be the mean height of NBA players (from sample).
   - \(\bar{x}_2\) be the mean height of NFL players (from sample).
   - The difference in sample means \((\bar{x}_1 - \bar{x}_2)\) will give us an estimate of the difference in population means.

2. **Calculate the Standard Error for the Difference in Means**:
   The standard error (SE) for the difference in two independent sample means is given by:
   \[
   SE = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}
   \]
   where:
   - \(s_1\) is the standard deviation of the NBA sample,
   - \(n_1\) is the size of the NBA sample,
   - \(s_2\) is the standard deviation of the NFL sample,
   - \(n_2\) is the size of the NFL sample.

3. **Find the Critical Value**:
   For a 92% confidence level, we need the critical value corresponding to the middle 92% of the normal distribution. This leaves 4% in the tails, or 2% in each tail. The critical value \(z\) for a 92% confidence level is approximately 1.75 (from the standard normal distribution).

4. **Calculate the Margin of Error (ME)**:
   The margin of error is calculated as:
   \[
   ME = z \times SE
   \]

5. **Determine the Confidence Interval**:
   The confidence interval for the difference in population means is:
   \[
   (\bar{x}_1 - \bar{x}_2) - ME < \mu_1 - \mu_2 < (\bar{x}_1 - \bar{x}_2) + ME
   \]

6. **Interpret the Confidence Interval**:
   - If the confidence interval contains only positive values, we can conclude that NBA players tend to be taller on average than NFL players.
   - If the interval contains only negative values, NFL players tend to be taller on average.
   - If the interval contains both negative and positive values, we cannot make a definitive statement about which group has a higher average height.

If you can provide the sample statistics for each group (means, standard deviations, and sample sizes), I can compute the exact values for margin of error, lower limit, and upper limit.

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Do you have the sample data for NBA and NFL players or need further guidance on any of these steps?

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### Follow-Up Questions

1. What is the importance of the confidence level in this context?
2. How would increasing the confidence level to 95% affect the interval?
3. What additional data might help make a stronger conclusion?
4. Can we use a similar method to compare other attributes of players, such as weight or experience?
5. What assumptions are necessary when calculating a confidence interval for mean difference?

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#### Tip:
When interpreting confidence intervals, remember that a larger interval implies more uncertainty about the estimate, while a narrower interval implies greater precision.

Two independent simple random samples of professional NBA and NFL players heights are given below. Find a 92% confidence interval for the difference in population means, rounding answers to two decimal places. Examine the confidence interval and explain what it means in the context of this problem.

Explore the calculation of a 92% confidence interval for the height difference between NBA and NFL players. This solution includes key statistical concepts, formulas, and interpretations of the results. Suitable for advanced high school students.

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