Question

Below is the PSPP analysis for ages (in years) of all 50 heads of households in a small Nova Scotian fishing village. Refer to the PSPP outputs below and answer the related questions.

Variable	N	Mean	Std Dev	Minimum	Maximum
Age	50	47.18	14.89	23.00	81.00

a) Interpret the standard deviation of age in the context of this study.

b) Suppose that we take a random sample of 36 heads of households. What would the sample data distribution tend to resemble more closely – the sampling distribution or the population distribution? Briefly Explain.

c) What do you expect for the mean of sample means in the long run of repeated samples of size 36?

d) What do you expect for the standard deviation of sample means in the long run of repeated samples of size 36? Show your work.

e) Explain the difference between a sample data distribution and the sampling distribution of sample mean.

f) Suppose that for a random sample of heads of households, we get a mean of 44.22, and standard deviation of 14.73. Before observing the sample, find the probability that our sample mean falls within 2.48 of the population mean. Interpret the result in the context of this problem.

Answer 1

(a)

Since the standard deviation of age = 14.89 is small, it tells us that the ages (in years) of all 50 heads of households in a small Nova Scotian fishing village. are not much spread out from mean = 47.18 years.

(b)

The sample data distribution tend to resemble more closely the population distribution because the sample data is drawn from the population.

(c)

We expect for the mean of sample means in the long run of repeated samples of size 36 to be equal to population mean = 47.18 years.

(d)

We expect for the standard deviation of sample means in the long run of repeated samples of size 36 to be equal to Standard Error given by:

(e)

The sample data distribution is the values of the variable for all the individuals in the sample. The sampling distribution of sample mean. is defined as follows: If repreated samples of a given size (here size = 36), are taken from a population with population mean (here 47.18) and population standard deviation (here 14.89) then the distribution of the mean of all sample means forms the sampling distribution of sample mean.

(f)

Z = 2.48/2.4817

= 1.00

Table of Area Under Standard Normal Curve gives area = 0.3413

So,

the probability that our sample mean falls within 2.48 of the population mean. = 2 X 0.3413 = 0.6826

So,

Answer is:

0.6826

Interpretation:

68.26% of sample means will fall within 2.48 of the population mean.