Question

A simple random sample of 60 items resulted in a sample mean of 97. The population standard deviation is 16.

a. Compute the 95% confidence interval for the population mean (to 1 decimal).

(  ,  )

b. Assume that the same sample mean was obtained from a sample of 120 items. Provide a 95% confidence interval for the population mean (to 2 decimals).

(  ,  )

c. What is the effect of a larger sample size on the margin of error?
SelectIt increasesIt decreasesIt stays the sameIt cannot be determined from the given dataItem 5

Answer 1

here we need to calculate 95% confidence interval for the given cases, in both cases population standard deviation is provided it means that some information about the population is given hence we will use z distribution to calculate 95% confidence interval using the following formula:

where zc is calculate the critical value using standard normal distribution at level of significance = alpha/2 i.e. 0.05/2 = 0.025.

n is the sample size and sigma is the population standard deviation.

Now Part a

Sample Mean Xbar=	97
Population Standard Deviation (σ) =	16
Sample Size (N) =	60

The critical value for α=0.05 is zc​=z1−α/2​=1.96.

(Here we can obtain this value using the standard normal distribution table

The corresponding confidence interval is computed as shown below:

Therefore, based on the data provided, the 95% confidence interval for the population mean is 92.952<μ<101.048, which indicates that we are 95\%95% confident that the true population mean μ is contained by the interval (92.952, 101.048).

Part 2

Now in this case sample size has increased to 120 so the calculation for 95% confidence interval will be:

Sample Mean Xbar =	97
Population Standard Deviation (σ) =	16
Sample Size (N)=	120

The critical value for α=0.05 is zc​=z1−α/2​=1.96.

(Here we can obtain this value using the standard normal distribution table

The corresponding confidence interval is computed as shown below:

Therefore, based on the data provided, the 95% confidence interval for the population mean is 94.137<μ<99.863, which indicates that we are %95% confident that the true population mean μ is contained by the interval(94.137,99.863).

Part 3

Effect of size on the margin of error

The margin of error when the sample size is 60:

= 4.048

The margin of error when the sample size is 120:

= 2.863

It can be seen that the sample size is in the denominator we can say that as the margin of error is larger sample size will be smaller

Hence we can say that there is an inverse relationship between the margin of error and sample size.

As the sample size increases, the margin of error decreases.