For a normally distributed population with a mean of 50 and a standard deviation of 10, when sampling with a sample size of 25, we can use the Central Limit Theorem.
The standard error of the mean (SEM) is calculated as the standard deviation of the population divided by the square root of the sample size:
SEM = σ / sqrt(n)
where σ is the population standard deviation and n is the sample size.
So, in this case:
SEM = 10 / sqrt(25) = 10 / 5 = 2
With a 95% confidence level, we use the z-score associated with 95% confidence, which is approximately 1.96 (from the standard normal distribution).
To find the range within which 95% of the sample means fall, we multiply the SEM by the z-score:
Range = SEM * z-score
Range = 2 * 1.96 = 3.92
So, we would expect 95% of the sample means to fall within ±3.92 units from the population mean of 50.
Therefore, the values between which 95% of the sample means fall are:
50 ± 3.92, or between approximately 46.08 and 53.92.
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