Assuming CLT applies for the sample data given by 54, 56, 59, 59, 65, 66, 67, 68, 69, 76, 84, 86, 89, 94, 101 Find a 90% CI to three decimals for the mean
To find the 90% confidence interval for the mean, we will use the formula:
CI = X̄ ± Z * (σ/√n),
where X̄ is the sample mean, Z is the z-score corresponding to the desired confidence level (90% in this case), σ is the population standard deviation (assuming it is known), and n is the sample size.
First, we calculate the sample mean (X̄) and the sample standard deviation (s):
X̄ = (54 + 56 + 59 + 59 + 65 + 66 + 67 + 68 + 69 + 76 + 84 + 86 + 89 + 94 + 101) / 15
= 994 / 15
≈ 66.267
Next, we calculate the sample standard deviation (s):
s = √((∑(Xi - X̄)²) / (n-1))
= √((∑(54-66.267)² + (56-66.267)² + (59-66.267)² + ... + (101-66.267)²)) / (15-1))
≈ √(13200.533) / 14
≈ √942.895
≈ 30.710
Assuming the population standard deviation is known, we can find the z-score corresponding to a 90% confidence level (α = 0.10 or 1-α = 0.90). Using a standard normal distribution table, we find the corresponding z-score to be approximately 1.645.
Finally, we can calculate the confidence interval:
CI = 66.267 ± 1.645 * (30.710/√15)
≈ 66.267 ± 1.645 * (30.710/√15)
≈ 66.267 ± 1.645 * (7.931)
≈ 66.267 ± 13.048
Therefore, the 90% confidence interval for the mean is approximately (53.219, 79.315) rounded to three decimals.