We are provided with the following data for the weight lost on a diet:
To find the confidence interval for the population mean μ\muμ, we first calculate the margin of error using the formula:
Margin of Error=z⋅sn \text{Margin of Error} = z \cdot \frac{s}{\sqrt{n}} Margin of Error=z⋅ns
For a 99%99\%99% confidence level, the critical value zzz is approximately 2.5762.5762.576. Thus, we compute:
Margin of Error=2.576⋅3.761≈2.576⋅0.473≈1.220 \text{Margin of Error} = 2.576 \cdot \frac{3.7}{\sqrt{61}} \approx 2.576 \cdot 0.473 \approx 1.220 Margin of Error=2.576⋅613.7≈2.576⋅0.473≈1.220
The confidence interval for the population mean μ\muμ is given by:
xˉ±Margin of Error \bar{x} \pm \text{Margin of Error} xˉ±Margin of Error
Substituting the values, we have:
20±1.220 20 \pm 1.220 20±1.220
This results in the confidence interval:
(20−1.220,20+1.220)=(18.780,21.220) (20 - 1.220, 20 + 1.220) = (18.780, 21.220) (20−1.220,20+1.220)=(18.780,21.220)
Rounding to one decimal place, we find:
Confidence Interval=(18.8,21.2) \text{Confidence Interval} = (18.8, 21.2) Confidence Interval=(18.8,21.2)
Since the sample size n=61n = 61n=61 is greater than 303030, we can apply the Central Limit Theorem, which states that the sampling distribution of the sample mean will be approximately normally distributed regardless of the population's distribution. Therefore, the confidence interval is not significantly affected by the fact that the population does not exhibit a normal distribution.
The confidence interval for the population mean μ\muμ is:
(18.8,21.2) \boxed{(18.8, 21.2)} (18.8,21.2)
The confidence interval is not affected by the population distribution because the sample size is large enough. Thus, the answer is:
C. No, because the sample size is large enough. \boxed{\text{C. No, because the sample size is large enough.}} C. No, because the sample size is large enough.
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