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# Sampling Error when Estimating a Population Proportion

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Assume the following data represents a population of 50 values. Values equal to 1 indicate that a particular attribute is present, a value equal to 0 indicates the attribute is not present.
1......1......1......1......1......1......1......1......0......0
0......1......1......0......0......1......1......1......0......1
0......1......1......1......1......1......1......1......1......0
1......1......1......1......1......1......1......0......1......1
0......1......1......1......1......1......1......1......1......1

a. Compute the population proportion.
The population proportionate is the fraction of values in a population, which have a specific attribute. Population proportionate = X number of items having the attribute divided by N, the population size.

b. A random sample of 15 items produced the following numbers: 1 1 1 0 0 1 0 0 1 1 0 0 0 1 0. Compute the sample proportion and the sampling error present in your sample.

c. What is the range of extreme sampling error for a sample of 15 taken from this population?

d. How would the range of extreme sampling error change if the sample size was set to 30? Discuss the advantages of having a larger sample size.

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Assume the following data represent a population of 50 values. Values equal
to 1 indicate that a particular attribute is present; a value equal to 0 indicates the attribute is not present.
1 1 1 1 1 1 1 1 0 0
0 1 1 0 0 1 1 1 0 1
0 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 0 1 1
0 1 1 1 1 1 1 1 1 1

a. Compute the population proportion.
The population proportion is the fraction of values in a population, which have a specific attribute. Population proportionate = X number of items having the attribute divided by N, the population size.

The attribute is present in 40 of the 50 members of the population. The population proportion having the attribute is:

b. A random sample of 15 items produced the following numbers: 1 1 1 0 0 1 0 0 1 1 0 0 0 1 0. Compute the sample proportion and the sampling error present in your sample.

The sample proportion is the fraction of the values in the sample, which have the attribute. The sample proportion is:

The sampling error is the difference between the estimate, , and the true parameter, p.

Sampling error:

c. What is the range of extreme sampling error for a sample of 15 taken from this population?

The standard error of is:

I will assume that the range of extreme sampling error is plus or minus 3 standard errors. The number of standard deviations could vary depending on your text and instructor. If you use a value other than 3 in your class, please substitute that value for 3 in what follows.

If the extreme sampling error is three times the standard error or , the error could occur either above or below the true value of p. The range, R, is therefore twice this amount or:

d. How would the range of extreme sampling error change if the sample size was set to 30? Discuss the advantages of having a larger sample size.

If the sample size, n, is changed to thirty, the standard error of changes to:

The range of extreme sampling error is this value times 6 or:

The point here is the following. The formula for the standard error of contains a ratio with n, the sample size, in the denominator. This means that as n increases, the standard error decreases. Since the range of extreme sampling error is a multiple of the standard error, it too decreases as the sample size increases.

The larger the sample size the smaller the variability of the estimate, , about the parameter, p, being estimated. The advantage of this is that you can make the standard error as small as you like by increasing the sample size.

I'm having problems working out C and D. Please explain step taken to arrive at answers.

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