Question

A study estimates the mean annual family income for families living in public housing in Chicago. For a random sample of 30 families, the annual incomes (in hundreds of dollars) are here:

133 140 127 150 133 114 128 142 123 172 146 110 135 136 158 120 189 106 144 134 161 143 170 120 142 150 174 109 162 129

(a) Produce appropriate plots and predict on the shape of the population distribu- tion?

(b) Find and interpret point estimates of μ and σ, the population mean and standard deviation.

(c) Construct and interpret a 95% confidence interval for μ.

can you please explain part a clearly

Answer 1

Answer:

Given that,

A study estimates the mean annual family income for families living in public housing in Chicago.

For a random sample of 30 families, the annual incomes (in hundreds of dollars) are here:

133 140 127 150 133 114 128 142 123 172 146 110 135 136 158 120 189 106 144 134 161 143 170 120 142 150 174 109 162 129

(a).

Produce appropriate plots and predict the shape of the population distribution:

Histogram of x:

Box plot of x:

Normal Q-Q plot:

By observing the graphs; we can expect the population distribution should be normal, Box plot supports the symmetry and from the qq-plot, we can see there is not a huge difference between the theoretical and observed deviation.

(b).

Find and interpret point estimates of μ and σ, the population means and standard deviation:

The parameters of the normal distributions are mean μ, and variance . Their estimates are the sample mean and sample variance S2.

Now,

and

So the estimate of

Mean gives the average value which we can expect. It is a measure of central tendency. For a normal distribution, most observations are clustered around the mean. Here it is 140.

Standard devition is the measure of dispersion. It gives average distance other observations are distributed away from mean. Here it is 20.6698.

(c).

Construct and interpret a 95% confidence interval for μ:

The equation for the confidence interval for the population mean is

We want to create the 95% confidence interval, i.e, .

From the t-tables with 29 df, we get =2.0452.

So the lower limit of the confidence interval is,

and the upper limit of the confidence interval is,

So the confidence interval is (132.2819,147.71810).