A researcher wishes to estimate the proportion of adults who have high-speed Internet access. What size sample should be obtained if she wishes the estimate to be within 0.03 with 90% confidence if (a) she uses a previous estimate of 0.34?(b) she does not use any prior estimates? n=_____(Round up to the nearest integer.)

Answer:

675 samples

752 samples

Step-by-step explanation:

Given that :

α = 90%

E = 0.03

Previous estimate (p) = 0.34

Estimated sample proportion (n)

n = p *q * (Zcritical /E)

Zcritical = α/2 = (1 - 0.9) /2 = 0.1 / 2 = 0.05

Z0.05 = 1.645 ( Z probability calculator)

q = 1 - p = 1 - 0.34 = 0.66

n = 0.34 * 0.66 * (1.645/0.03)^2

n = 674.70

n = 675

B.) without prior estimate given ;

p and q should have equal probability

Hence ;

p = 0.5 ; q = 0.5

n = 0.5 * 0.5 * (1.645/0.03)^2

n = 751.67361

n = 752

Final answer:

To determine the sample size for the research, a mathematical calculation is required, considering the level of confidence, the desired margin of error, and, if available, a previous estimate. For a 90% confidence level, two calculations are made based on an assumed proportion—first using the existing estimate of 0.34, and second with no prior estimate (commonly taken as 0.5).

Explanation:

The researcher is trying to determine the sample size required for a study on high-speed Internet access. The sample size can be calculated based on the level of confidence desired and the desired margin of error. In this case, (a) she is using a previous estimate of 0.34 and (b) she does not use any prior estimates.

For the calculation, we can use the formula for sample size in hypothesis testing for a population proportion:

n = (Z^2 * p * (1-p)) / E^2

Where:

• n = required sample size
• Z = z-score corresponding to our desired level of confidence
• p = assumed proportion (in part (a) this is 0.34)
• E = desired margin of error

For a 90% confidence level, the corresponding z-score is approximately 1.645. Now we can plug in the numbers.

(a) If she uses a previous estimate of 0.34, the calculation would be:

n = (1.645^2 * 0.34 * 0.66) / (0.03)^2

You will need to round up to the nearest whole number as you cannot have a part of a participant.

(b) If no previous estimate is used, it is common practice to use 0.5 as this will provide the maximum variance and, therefore, the largest sample size. The calculation would be:

n = (1.645^2 * 0.5 * 0.5) / (0.03)^2

Again, round up to the next whole number. Insert the results for these calculations to your answer.

Learn more about Sample Size Calculation here:

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