You obtain a sample mean of 75 from a population with a known mean of 70 and standard deviation of 14. If your sample size is 49, what is the z-score for your obtained mean?

Question

BlackTom AI · Accepted Answer

First, identify the given values: population mean mu = 70, population standard deviation sigma = 14, sample size n = 49, and the sample mean x̄ = 75.
Next, recall the formula for the z-score of a sample mean: z = (x̄ - mu) / (sigma / sqrt(n)).
Compute the standard error of the mean: sigma / sqrt(n) = 14 / sqrt(49) = 14 / 7 = 2.
Now plug the values into the formula: z = (75 - 70) / 2 = 5 / 2 = 2.5.
Interpretation note: a z-score of 2.5 indicates the observed sample mean is 2.5 standard errors above the population mean, assuming the population standard deviation is known and the sampling distribution is normal (which is justified here by a reasonably large n and the use of sigma).
Potential misunderstandings to avoid: using the sample standard deviation in the denominator (which would be s / sqrt(n) and would produce a different value), or forgetting to divide by sqrt(n) when computing the standard error, or using the wrong order of subtraction (x̄ - mu) versus (mu - x̄).
If the question had asked for a two-tailed p-value, this z of 2.5 would correspond to a small but non-negligible tail probability, but since only the z-score is requested, the primary result is 2.5.

You obtain a sample mean of 75 from a population with a known mean of 70 and standard deviation of 14. If your sample size is 49, what is the z-score for your obtained mean? Numerical

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