To construct a 98% confidence interval for the population mean, we can use the formula:
CI = x̄ ± Z * (σ/√n)
Where:
CI = Confidence Interval
x̄ = Sample mean
Z = Z-score (from the Z distribution)
σ = Population standard deviation
n = Sample size
Since we are given the sample variance (20.25) and the sample mean (32), we can use the sample standard deviation as an estimate for the population standard deviation. The sample standard deviation is the square root of the sample variance, which is √20.25 = 4.5.
Given that we want a 98% confidence interval, the Z-score corresponding to a 98% confidence level is 2.33 (from the standard normal distribution).
Plugging in the values, we get:
CI = 32 ± 2.33 * (4.5/√81)
CI = 32 ± 2.33 * 0.5
CI = 32 ± 1.165
Therefore, the 98% confidence interval for the population mean is (30.835, 33.165).
A random sample of size 81 was taken whose variance is 20.25 and means 32 construct 98% confidence limits
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