a recent poll of 700 peole who work indoors found that 278 of the smoke. If the researchers want to be 98% confident of thier results to within 3.5% how large a sample is necessary?
how do i work this problem?
You are correct
Try this formula:
n = [(z-value)^2 * p * q]/E^2
= [(2.33)^2 * .397 * .603]/.035^2
I'll let you finish the calculation.
Note: n = sample size needed; .397 (which is approximately 278/700 in decimal form) for p and .603 (which is 1 - p) for q. E = maximum error, which is .035 (3.5%) in the problem. Z-value is found using a z-table (for 98%, the value is approximately 2.33). Symbols: * means to multiply and ^2 means squared.
I hope this will help.
To determine the necessary sample size, you can use the formula for calculating the sample size for estimating a proportion.
The formula is:
n = [z^2 * p * (1-p)] / E^2
Where:
n = sample size
z = Z-score for the desired level of confidence (use a Z-score table or calculator)
p = estimated proportion (in this case, 278/700 = 0.397)
E = maximum desired margin of error (3.5% of the estimated proportion = 0.035)
Substituting the values into the formula:
n = [(Z^2 * p * (1-p)) / E^2]
Now, let's calculate the sample size using a Z-score of 2.33 for a 98% confidence level:
n = [(2.33^2 * 0.397 * (1-0.397)) / 0.035^2]
n = [(5.4289 * 0.397 * 0.603) / 0.001225]
n = (1.02793923) / 0.001225
n ≈ 838.715
Therefore, a sample size of approximately 839 is necessary to be 98% confident of the results within a margin of error of 3.5%.
To determine the sample size necessary to achieve the desired level of confidence and margin of error, you can use the formula for sample size in estimating proportions. Here's how you can work through the problem:
1. Determine the level of confidence: In this case, the researchers want to be 98% confident, so the level of confidence (C) is 0.98.
2. Find the margin of error: The margin of error (E) is the desired maximum difference or precision allowed from the estimated result. In this case, it is stated as within 3.5%, which can be expressed as 0.035.
3. Calculate the standard deviation: To determine the standard deviation (σ), you need to estimate the proportion of people who smoke indoors. This is given as 278 out of 700 people, so the estimated proportion (p) is 278/700 = 0.397 (to decimal precision).
4. Compute the critical value: The critical value (Z) corresponds to the desired level of confidence. For a 98% confidence level, you can look up the value using a Z-table or use a statistical calculator/tool, which will give you Z = 2.326.
5. Apply the formula: Now you can use the formula for sample size:
n = (Z^2 * p * (1 - p)) / E^2
Substitute the values you've calculated:
n = (2.326^2 * 0.397 * (1 - 0.397)) / 0.035^2
6. Calculate the sample size: Now, plug the values into a calculator or simplify the expression to find the sample size required. The result will be the minimum sample size needed to achieve the desired level of confidence and margin of error.
n ≈ 274.52
Since you can't have a fraction of a person in your sample, you would round up to the nearest whole number.
Therefore, a sample size of 275 or more is necessary to achieve a 98% confidence level with a margin of error of 3.5%.