Use the given information to find the minimum sample size required to estimate an unknown population mean. How many business students must be randomly selected to estimate the mean monthly earnings of business students at one college? We want a 95% confidence that the sample mean is within $128 of the population mean, and the population standard deviation is known to be $536.
To find the minimum sample size required to estimate an unknown population mean, we can use the formula:
n = (Z * σ / E)^2
where:
n = sample size
Z = Z-score corresponding to the desired level of confidence (in this case 95%)
σ = population standard deviation
E = margin of error
Given:
Z = 1
To find the minimum sample size required to estimate an unknown population mean, we can use the formula:
n = (Z * σ / E)^2
where:
n = sample size
Z = Z-score corresponding to the desired level of confidence (in this case 95%)
σ = population standard deviation
E = margin of error
Given:
Z = 1.96 (for a 95% confidence level, as it corresponds to 2 standard deviations)
σ = $536
E = $128
Plugging in these values into the formula:
n = (1.96 * 536 / 128)^2
Calculating this expression:
n = (1050.56 / 128)^2
n = (8.22)^2
n ≈ 67.47
Since the sample size must be a whole number, we round up to the next nearest whole number.
Therefore, a minimum sample size of approximately 68 business students must be randomly selected to estimate the mean monthly earnings of business students at one college, in order to have a 95% confidence that the sample mean is within $128 of the population mean.
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