Answer :
The margin of error for a sample of size 49, using the 99.7% part of the 68-95-99.7 rule, is approximately ±24.77.
The margin of error for a sample of size 49, using the 95% part of the 68-95-99.7 rule, is approximately ±12.11.
To compute the margin of error, we need to calculate the standard error. The standard error is the standard deviation of the sampling distribution, which is equal to the population standard deviation divided by the square root of the sample size.
For the 99.7% confidence level, we consider the two tails of the distribution, which correspond to 0.15% each. Since we have a symmetric distribution, we divide this probability by 2 to get the tail probability of 0.15%. Using a Z-table or calculator, we find that the Z-score corresponding to a tail probability of 0.15% is approximately 2.97.
For the 95% confidence level, we consider the two tails of the distribution, which correspond to 2.5% each. Again, since we have a symmetric distribution, we divide this probability by 2 to get the tail probability of 2.5%. Using a Z-table or calculator, we find that the Z-score corresponding to a tail probability of 2.5% is approximately 1.96.
Now, we can calculate the standard error for both cases:
Standard error (99.7%) = 84 / √49 ≈ 12.00
Standard error (95%) = 84 / √49 ≈ 12.00
Finally, we multiply the standard error by the respective Z-score to obtain the margin of error:
Margin of error (99.7%) ≈ 2.97 * 12.00 ≈ 35.64 ≈ ±24.77
Margin of error (95%) ≈ 1.96 * 12.00 ≈ 23.52 ≈ ±12.11
For a sample size of 49, the margin of error is approximately ±24.77 for a 99.7% confidence level and ±12.11 for a 95% confidence level. This means that if we take repeated samples and construct confidence intervals using these samples, we would expect the true population mean to fall within these ranges in 99.7% and 95% of the cases, respectively. The larger margin of error for the 99.7% confidence level reflects a wider interval due to the higher level of confidence required. Conversely, the smaller margin of error for the 95% confidence level corresponds to a narrower interval since less confidence is required.
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