- Find the sample size needed for a sampling distribution to have a desired standard deviation.
- Check the conditions for normal approximation of a sampling distribution of a sample proportion.
- Use the normal distribution to calculate probabilities and percentiles from a sampling distribution.
Calculating the Sample Size [latex]n[/latex]
If researchers desire a specific standard error, then they can use the standard error formula to calculate the required sample size.
[latex]SE = \sqrt{\dfrac{p(1-p)}{n}}[/latex]
[latex]SE^2 = \dfrac{p(1-p)}{n}[/latex]
[latex]n \times SE^2 = p(1-p)[/latex]
[latex]n = \dfrac{p(1-p)}{SE^2}[/latex]
How many customers aged 50+ should the company survey in order to be [latex]68\%[/latex] confident that the estimated (sample) proportion is within three percentage points of the true population proportion of customers aged 50+ who use text messaging on their cell phones? Answer: [latex]278[/latex] customers.
Explanation: Recall that [latex]68\%[/latex] of the data falls within [latex]1[/latex] standard deviation from the mean. This means that [latex]3\%[/latex] is the standard deviation (or the standard error) desired.
[latex]n = \dfrac{p(1-p)}{SE^2}[/latex]
However, in order to find [latex]n[/latex], we need to know the estimated (sample) proportion [latex]p[/latex]. But, we do not know [latex]p[/latex] yet.
Since we multiply [latex]p[/latex] and [latex](1-p)[/latex] together, we make [latex]p[/latex] equal to [latex]0.5[/latex] because [latex]p(1-p)=(0.5)(0.5) = 0.25[/latex] results in the largest possible product. (Try other product: [latex]p(1-p)=(0.6)(0.4) = 0.24[/latex], [latex]p(1-p)=(0.7)(0.3) = 0.21[/latex], and so on.)
[latex]n = \dfrac{p(1-p)}{SE^2}[/latex]
[latex]n = \dfrac{0.5(1-0.5)}{0.03^2}[/latex]
[latex]n \approx 277.78[/latex]
Round the answer to the next higher value.
The sample size should be [latex]278[/latex] cell phone customers aged 50+ in order to be [latex]68\%[/latex] confident that the estimated (sample) proportion is within three percentage points of the true population proportion of all customers aged 50+ who use text messaging on their cell phones.