SOLUTION: A government official is in charge of allocating social programs throughout the city of Vancouver. He will decide where these social outreach programs should be located based on th

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Question 1201682: A government official is in charge of allocating social programs throughout the city of Vancouver. He will decide where these social outreach programs should be located based on the percentage of residents living below the poverty line in each region of the city. He takes a simple random sample of 124 people living in Gastown and finds that 20 have an annual income that is below the poverty line.
Part i) The proportion of the 124 people who are living below the poverty line, 20/124, is a:
A. parameter.
B. statistic.
C. variable of interest.
Part ii) Use the sample data to compute a 95% confidence interval for the true proportion of Gastown residents living below the poverty line.
(Please carry answers to at least six decimal places in intermediate steps. Give your final answer to the nearest three decimal places).
95% confidence interval = ( , )
Answer by math_tutor2020(3817) (Show Source):
You can put this solution on YOUR website!

Part (i)

Parameters measure an aspect about the population (both start with P).
parameter -> population

Statistics measure an aspect about a sample (both start with s).
statistic -> sample

Example:
xbar is the sample mean (statistic)
and it estimates the population mean mu (parameter)

The 124 refers to a sample size.
The 20/124 is the sample proportion (phat)

Answer: B. Statistic

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Part (ii)

p = population proportion = parameter
phat = sample proportion = statistic

The job of phat is to estimate p.
It's called "phat" because the letter p has a little hat on top.
phat =

At 95% confidence, the z critical value is roughly z = 1.960
Use a table like this
https://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf
to get that value. Look at the bottom row labeled "Z" and above the 95% confidence level.
A table similar to this should be in the back of your textbook.
A stats calculator can also compute this value.

x = number of people below the poverty line = 20
n = sample size = 124
phat = sample proportion
phat = x/n
phat = 20/124
phat = 0.1612903 approximately

E = margin of error for a proportion
E = z*sqrt(phat*(1-phat)/n)
E = 1.960*sqrt(0.1612903*(1-0.1612903)/124)
E = 0.0647374

L = lower boundary of confidence interval
L = phat - E
L = 0.1612903 - 0.0647374
L = 0.0965529
L = 0.097
and
U = upper boundary of confidence interval
U = phat + E
U = 0.1612903 + 0.0647374
U = 0.2260277
U = 0.226

The 95% confidence interval in the format
L < p < U
is approximately
0.097 < p < 0.226
and that condenses to
(0.097, 0.226)
I prefer the 1st format mentioned because it tells us what parameter we're trying to estimate.
But most textbooks and other settings will use the shortened 2nd format more often.

The interpretation would be "we are 95% confident the population proportion of those below the poverty line is somewhere between 0.097 and 0.226"
0.097 = 9.7%
0.226 = 22.6%

Answer: (0.097, 0.226)