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Sagot :
Sure, let's go through this step-by-step to construct a 90% confidence interval for the given data and understand what the result tells us about the population.
### Steps to Construct the Confidence Interval:
1. Data Collection and Preliminary Calculations:
[tex]\[ \text{The given data: } [258, 199, 199, 168, 161, 153, 148, 148, 143, 143] \][/tex]
2. Calculate the Sample Mean ([tex]\( \bar{x} \)[/tex]):
[tex]\[ \bar{x} = 172.0 \text{ million} \][/tex]
3. Calculate the Sample Standard Deviation (s):
[tex]\[ s = 36.8 \text{ million} \quad \text{(rounded to one decimal place)} \][/tex]
4. Sample Size (n):
[tex]\[ n = 10 \][/tex]
5. Determine the Confidence Level:
[tex]\[ \text{Confidence Level} = 90\% \][/tex]
6. Find the Z Critical Value ( [tex]\(z_{\alpha/2} \)[/tex] ):
The critical value for a 90% confidence interval is approximately [tex]\( z_{\alpha/2} = 1.645 \)[/tex].
7. Calculate the Margin of Error (ME):
[tex]\[ ME = z_{\alpha/2} \times \left( \frac{s}{\sqrt{n}} \right) \][/tex]
[tex]\[ ME = 1.645 \times \left( \frac{36.8}{\sqrt{10}} \right) \][/tex]
[tex]\[ ME = 19.1 \text{ million} \quad \text{(rounded to one decimal place)} \][/tex]
8. Calculate the Confidence Interval:
[tex]\[ (\bar{x} - ME, \bar{x} + ME) \][/tex]
[tex]\[ (172.0 - 19.1, 172.0 + 19.1) \][/tex]
[tex]\[ (152.9, 191.1) \text{ million} \][/tex]
So, the 90% confidence interval estimate for the population mean [tex]\(\mu\)[/tex] is:
[tex]\[ \$152.9 \text{ million} < \mu < \$191.1 \text{ million} \][/tex]
### Interpretation:
- The 90% confidence interval for the mean net worth of the wealthiest celebrities in the country is between \[tex]$152.9 million and \$[/tex]191.1 million. This means we are 90% confident that the true population mean net worth of all celebrities in the country lies within this interval.
- Whether the data is from a normally distributed population is a judgment based on the provided data. Given the relatively small sample size of 10, it's difficult to confirm normality without additional information such as a normality test or Q-Q plot. However, with a larger sample size, the Central Limit Theorem suggests that the sampling distribution of the mean would be approximately normal, which could make the confidence interval estimates valid.
In summary, based on the given data and calculations, the 90% confidence interval estimate for the population mean net worth of the celebrities is:
[tex]\[ \$152.9 \text{ million} < \mu < \$191.1 \text{ million} \][/tex]
rounded to one decimal place as needed.
### Steps to Construct the Confidence Interval:
1. Data Collection and Preliminary Calculations:
[tex]\[ \text{The given data: } [258, 199, 199, 168, 161, 153, 148, 148, 143, 143] \][/tex]
2. Calculate the Sample Mean ([tex]\( \bar{x} \)[/tex]):
[tex]\[ \bar{x} = 172.0 \text{ million} \][/tex]
3. Calculate the Sample Standard Deviation (s):
[tex]\[ s = 36.8 \text{ million} \quad \text{(rounded to one decimal place)} \][/tex]
4. Sample Size (n):
[tex]\[ n = 10 \][/tex]
5. Determine the Confidence Level:
[tex]\[ \text{Confidence Level} = 90\% \][/tex]
6. Find the Z Critical Value ( [tex]\(z_{\alpha/2} \)[/tex] ):
The critical value for a 90% confidence interval is approximately [tex]\( z_{\alpha/2} = 1.645 \)[/tex].
7. Calculate the Margin of Error (ME):
[tex]\[ ME = z_{\alpha/2} \times \left( \frac{s}{\sqrt{n}} \right) \][/tex]
[tex]\[ ME = 1.645 \times \left( \frac{36.8}{\sqrt{10}} \right) \][/tex]
[tex]\[ ME = 19.1 \text{ million} \quad \text{(rounded to one decimal place)} \][/tex]
8. Calculate the Confidence Interval:
[tex]\[ (\bar{x} - ME, \bar{x} + ME) \][/tex]
[tex]\[ (172.0 - 19.1, 172.0 + 19.1) \][/tex]
[tex]\[ (152.9, 191.1) \text{ million} \][/tex]
So, the 90% confidence interval estimate for the population mean [tex]\(\mu\)[/tex] is:
[tex]\[ \$152.9 \text{ million} < \mu < \$191.1 \text{ million} \][/tex]
### Interpretation:
- The 90% confidence interval for the mean net worth of the wealthiest celebrities in the country is between \[tex]$152.9 million and \$[/tex]191.1 million. This means we are 90% confident that the true population mean net worth of all celebrities in the country lies within this interval.
- Whether the data is from a normally distributed population is a judgment based on the provided data. Given the relatively small sample size of 10, it's difficult to confirm normality without additional information such as a normality test or Q-Q plot. However, with a larger sample size, the Central Limit Theorem suggests that the sampling distribution of the mean would be approximately normal, which could make the confidence interval estimates valid.
In summary, based on the given data and calculations, the 90% confidence interval estimate for the population mean net worth of the celebrities is:
[tex]\[ \$152.9 \text{ million} < \mu < \$191.1 \text{ million} \][/tex]
rounded to one decimal place as needed.
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