Respuesta :
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 [tex]\bar X[/tex] = sample mean
The z-score probability distribution for a sample mean is given by;
Z = [tex]\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }[/tex] ~ N(0,1)
where, [tex]\mu[/tex] = population mean = 70
[tex]\sigma[/tex] = 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 = [tex]\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }[/tex] = [tex]\frac{\bar X-70}{\frac{8}{\sqrt{4} } }[/tex]
2.00 = [tex]\frac{\bar X-70}{\frac{8}{\sqrt{4} } }[/tex]
2.00 = [tex]\frac{\bar X-70}{4 } }[/tex]
[tex]\bar X = 70+(2 \times 4)[/tex]
[tex]\bar X[/tex] = 70 + 8 = 78
Therefore, the value of the sample mean is 78.
