Question

A sample of n = 4 scores is obtained from a population with a mean of 70 and a standard deviation of 8. If the sample mean corresponds to a z score of 2.00, then what is the value of the sample mean?

Answer 1

Answer:

The value of the sample mean is 78.

Step-by-step explanation:

We are given that a sample of n = 4 scores is obtained from a population with a mean of 70 and a standard deviation of 8.

Also, the sample mean corresponds to a z score of 2.00.

Let $\bar X$ = sample mean

The z-score probability distribution for a sample mean is given by;

              Z = $\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }$  ~ N(0,1)

where, $\mu$ = population mean = 70

            $\sigma$ = standard deviation = 8

            n = sample size = 4

The Z-score measures how many standard deviations the measure is away from the mean. After finding the Z-score, we look at the z-score table and find the p-value (area) associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X.

Now, we are given that the sample mean corresponds to a z score of 2.00 for which we have to find the value of sample mean;

So, z-score formula is given by ;

                  z-score = $\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }$ = $\frac{\bar X-70}{\frac{8}{\sqrt{4} } }$

                      2.00 =  $\frac{\bar X-70}{\frac{8}{\sqrt{4} } }$

                      2.00 =  $\frac{\bar X-70}{4 } }$

                     $\bar X = 70+(2 \times 4)$

                     $\bar X$ = 70 + 8 = 78

Therefore, the value of the sample mean is 78.