13. a population forms a normal distribution with a mean of $mu = 85$ and a standard deviation of $sigma =…

13. a population forms a normal distribution with a mean of $mu = 85$ and a standard deviation of $sigma = 24$. for each of the following samples, compute the $z$-score for the sample mean.\na. $m = 91$ for $n = 4$ scores\nb. $m = 91$ for $n = 9$ scores\nc. $m = 91$ for $n = 16$ scores\nd. $m = 91$ for $n = 36$ scores

13. a population forms a normal distribution with a mean of $mu = 85$ and a standard deviation of $sigma = 24$. for each of the following samples, compute the $z$-score for the sample mean.\na. $m = 91$ for $n = 4$ scores\nb. $m = 91$ for $n = 9$ scores\nc. $m = 91$ for $n = 16$ scores\nd. $m = 91$ for $n = 36$ scores

Answer

Explanation:

Step1: Recall the formula for the z - score of the sample mean

The formula for the z - score of the sample mean is (z=\frac{M-\mu}{\frac{\sigma}{\sqrt{n}}}), where (M) is the sample mean, (\mu) is the population mean, (\sigma) is the population standard deviation, and (n) is the sample size.

Step2: Substitute the values for part a

Given (\mu = 85), (\sigma=24), (M = 91), (n = 4) First, calculate (\frac{\sigma}{\sqrt{n}}=\frac{24}{\sqrt{4}}=\frac{24}{2}=12) Then, (z=\frac{91 - 85}{12}=\frac{6}{12}=0.5)

Step3: Substitute the values for part b

Given (\mu = 85), (\sigma=24), (M = 91), (n = 9) First, calculate (\frac{\sigma}{\sqrt{n}}=\frac{24}{\sqrt{9}}=\frac{24}{3}=8) Then, (z=\frac{91 - 85}{8}=\frac{6}{8}=0.75)

Step4: Substitute the values for part c

Given (\mu = 85), (\sigma=24), (M = 91), (n = 16) First, calculate (\frac{\sigma}{\sqrt{n}}=\frac{24}{\sqrt{16}}=\frac{24}{4}=6) Then, (z=\frac{91 - 85}{6}=\frac{6}{6}=1)

Step5: Substitute the values for part d

Given (\mu = 85), (\sigma=24), (M = 91), (n = 36) First, calculate (\frac{\sigma}{\sqrt{n}}=\frac{24}{\sqrt{36}}=\frac{24}{6}=4) Then, (z=\frac{91 - 85}{4}=\frac{6}{4}=1.5)

Answer:

a. (z = 0.5) b. (z=0.75) c. (z = 1) d. (z = 1.5)