Question

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 129 people living in Gastown and finds that 21 have an annual income that is below the poverty line.

Part i) The proportion of the 129 people who are living below the poverty line, 21/129, 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 (your answer should be a proportion).

(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 1

Here we have to use the given information to answer two questions.

Part i)

Here, as we have taken a random sample of 129 people living in Gastown, so the Population Proportion, P will be the Parameter and the sample proportion (21/129) , obtained from our random sample will be our Statistic.

Therefore,

The proportion of the 129 people who are living below the poverty line, 21/129, is a:

A. parameter.
B. statistic.
C. variable of interest.

Hence, B. statistic is the correct alternative.

Now, let us consider the part two.

Part ii)

Here, we have to construct a 95% Confidence Interval for the True Proportion ( i.e. Population Proportion) of Gastown residents.

We are given that,

Sample size, n = 129

p= 21/129

q=(1-p)= 108/129

Hence,

Further, a 95% Confidence Interval for the True Proportion, P is given as :-

Where, p^ is the estimate of P.

Putting the values, we get :-

Hence, a 95% Confidence Interval for True Proportion is

But remember that, Proportion lies between (0,1) , so it cannot be negative.

Hence, our confidence interval reduces to

This is the required 95% Confidence Interval for the True Proportion of the Gastown residents.

This answers your question.

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