Question

The amounts of time employees at a large corporation work each day are normally distributed, with a mean of 7.5 hours and a standard deviation of 0.35 hour. Random samples of size 23 and 35 are drawn from the population and the mean of each sample is determined. What happens to the mean and the standard deviation of the distribution of sample means as the size of the sample increases?a) If the sample size is n=23, ind the mean and standard deviation of the distribution of sample means (type as interger or a decimal)b)If the sample size is n=35, find the mean and standard dviation of the sidtribution of sample meansean(type as interger or a decimal)=standard deviation(round to the nearest hundredth)=c) The mean (stays the same/and the standard deviation) (both decrease/both increase/ the standard deviation increase/ the standard deviation decreases)

Answer 1

When sample size increases from n = 23 to n = 35, the mean of the distribution of sample means remains same but the standard deviation of the distribution of sample means decreases from 0.073 to 0.059.

Explanation:

We are given the amounts of time employees at a large corporation work each day are normally distributed, with a mean of 7.5 hours and a standard deviation of 0.35 hour.

Random samples of size 23 and 35 are drawn from the population and the mean of each sample is determined.

Here,
`\mu` = population mean = 7.5 hours

`\sigma` = population standard deviation = 0.35 hour

Let
`\bar X` = sample mean

(a) The random samples of size of n = 23 is drawn from the population;

So, mean of the distribution of sample means =
`\mu` = 7.5 hours

Standard deviation for the distribution of sample means is given by;

s =
`(\sigma)/(√(n) )` =
`(0.35)/(23)` = 0.073

(b) The random samples of size of n = 35 is drawn from the population;

So, mean of the distribution of sample means =
`\mu` = 7.5 hours

Standard deviation for the distribution of sample means is given by;

s =
`(\sigma)/(√(n) )` =
`} (0.35)/(√(35) )` = 0.059

(c) So, as we can see that when sample size increases from n = 23 to n = 35, the mean of the distribution of sample means remains same but the standard deviation of the distribution of sample means decreases from 0.073 to 0.059.