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Last Name 1 Student’s Name Professor’s Name Course Number Date Part 2: V3, Quantitative Data A. Age at death B. C. Mean= 𝜇 = 74.625 Standard deviation= 𝜎 = 9.9 Data Group , Using V3 f frequency Class Midpoint x x. f 40-49 1 45 45 50-59 2 55 110 60-69 7 65 455 70-79 16 75 1200 80-89 11 85 935 90-99 3 95 285 Total 40 3030

Last Name 2 PART 1: Confidence Interval for Quantitative Data, V3 A. Given: Sample mean ( x̄ ) = 74.625 Population standard deviation (σ) = 9.9 Sample size (n) = 40 95% Confidence Interval: The formula for the confidence interval for the population mean ( μ ) when the population standard deviation ( σ ) is known is: Confidence Interval= 𝑥̅ ± 𝑍 ∗ 𝜎 √ 𝑛 For a 95% confidence interval, the Z-score is 1.96 (from the standard normal distribution table). Confidence Interval= 74 .625 ± 1 .96 ∗ 9 .9 √ 40 Confidence Interval= 74 .625 ± 3 .068 95% Confidence interval= (71.557, 77.693) B. I am 95% confident that the actual mean age at death of the population is between 71.557 and 77.693 years. C. A. For a 99% confidence interval, the Z-score is 2.576 (from the standard normal distribution table). Confidence Interval= 74 .625 ± 2 .576 ∗ 9 .9 √ 40

Last Name 3 Confidence Interval= 74 .625 ± 4 .032 99% Confidence interval= (70.593, 78.657) B. I am 99% confident that the actual mean age at death of the population is between 70.593 and 78.657 years. D. The 99% confidence interval is wider than the 95% confidence interval. This difference occurs because higher confidence levels require a wider range to capture the true population mean. In other words, with a higher confidence level (99% compared to 95%), we are more certain about our estimation, so the interval needs to be wider to accommodate a larger range of potential values for the population mean. PART 2: Sample Size Determination A. Desirable Value for E (Margin of Error): Selected E (margin of error) = 1 year B. The formula to determine the minimum sample size required to achieve a specific margin of error (E) with a 95% degree of confidence is: n= ( 𝑍 ∗𝜎 𝐸 ) 2

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Last Name 4 Given that E =1 year, Z =1.96 (for a 95% confidence level), and σ =9.9 (approximated as sample standard deviation): n= ( 1 .96 ∗9 .9 1 ) 2 n= ( 19 .404) 2 n= 376.52 Rounding up to nearest whole number: n= 377 Therefore, the minimum sample size required to produce results accurate to a 95% degree of confidence and a margin of error of 1 year is 377. C. With a sample size of at least 177, I will be 95% confident that my sample mean will be within 1.5 years of the true mean age at death.

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