2. Proportion Never Married A sampling distribution is shown for the proportion

Question

2. Proportion Never Married A sampling distribution is shown for the proportion of US citizens over 15 years old who have never been married, using the data from the 2010 US Census and random samples of size n - 500 0.26 0.28 0.30 0.32 0.34 0.36 0.38 a). What does one dot in the dotplot represent? Use proper grammar and sentence structure when stating your response! b). Use the sampling distribution to estimate the proportion of all US citizens over 15 years old who have never been married. Give correct notation for your answer c). Estimate the standard error of the sampling distributiorn d). If we took samples of size 1000 instead of 500, and used the sample proportions to estimate the population proportion Would the estimates be more accurate or less accurate? Would the standard error be larger or smaller?

Explanation / Answer

(a)

One dot in the dotplot is the sample proportion value for the random sample of size n = 500. It represents the proportion of US citizens over 15 years old who have never been married for that sample.

(b)

Since a sampling distribution for any population is always very close to normal, so the mean value is the exact centre. Looking at the dotplot, the expected value of the sampling distribution is 0.32

(c)

Standard error of sampling distribution is another name for its standard deviation.

Since 0.32 is the mean value, and from the empirical rule we know that almost 99.7% values for a normal distribution lie within 3 Standard deviations of mean, so the length of three standard deviations to the right of the mean is equal to: (0.38-0.32) = 0.06

So,

Standard deviation = 0.06/3 = 0.02

(d)

The estimates would be more accurate if we use large sample sizes, but the mean value of the sampling distribution would be the same no matter what sample size you choose.

Standard error is inversely proportional to the square root of sample size, so when the sample size is doubled, from 500 to 1000, then the standard error would be smaller.

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