The percentage of the Iowa voting-eligible population that voted in the February

Question

The percentage of the Iowa voting-eligible population that voted in the February 1 presidential primary election was 15.7% (Source: http://www.electproject.org/). Suppose we take a simple random sample of 50 people from the Iowa voting-eligible population. Let X be the number of these 50 people that voted in the February 1 presidential primary election, and ˆp = X/50 be the proportion of these 50 people that voted in the Feburary 1 presidential primary election.

(a) Is the number 15.7% a parameter or a statistic? Why?

(b) Is ˆp a parameter or a statistic? Why?

(c) Is ˆp a discrete or continuous random variable? Why?

(d) What is the mean of the sampling distribution of X? of ˆp?

(e) What is the standard deviation of the sampling distribution of X? of ˆp?

(f) Using the applet from discussion activity #4, simulate 500 simple random samples of 50 people from this population (uncheck the Animate box!). Select “Number of orange” and check the “Summary Stats” box. Turn in a screen shot of your simulation.

i. What are the simulated mean and standard deviation of the distribution? How do these compare to the mean and standard deviation of X from parts (d) and (e)?

ii. Select “Proportion of orange”. What changed in your plot? Explain why the shape of the distribution remained the same.

(g) Use R Commander to calculate P(ˆp 0.2) using the binomial distribution. Use R Commander to calculate P(ˆp 0.2) using the normal distribution. Estimate P(ˆp 0.2) using your simulation from part (f). How do these three values compare to each other? Which value is the most accurate?

(h) Suppose a news agency polls a simple random sample of 50 voter-eligible Iowans and pˆ = 0.20. Use this sample result to calculate a 95% confidence interval for the proportion of voter-eligible Iowans that voted in the February 1 presidential primary election. Did this interval capture the parameter it was trying to estimate?

Explanation / Answer

(a)The number 15.7% is a parameter. Because this value describes the whole population. In other words, a parameter is a numerical quantity which characterizes a given population or some aspect of it.

(b) ˆp is a sample statistic. A statistic is a piece of data sampled from a population. Statistics describe the data and the variable.

(c) ˆp is a discrete random variable.It takes distinct values 1,2,…

(d)The mean is same as the population mean.

(e)The standard deviation is:

p^(1-p^)/n

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