You want to make a 97 % 97\% 97% confidence interval for the population proportion of people between 20 − 30 20-30 20 − 30 years old who have gotten a speeding ticket in the past 2 2 2 years. A prior study found that 26 % 26\% 26% of people between 20 − 30 20-30 20 − 30 years old have received a speeding ticket in the last year. If you want your estimate to be accurate within 3 % 3\% 3% of the true population proportion, what is the minimum sample size needed?

wanna make a 97% confidence interval for the population proportion of people between 20 to 30 years old who've gotten a speeding ticket in the past two years. Now a prior study found that twenty six percent of people between those ages have received a speeding ticket in those last two years. If you want your estimate to be accurate within three percent of the true population proportion, what is the minimum sample size needed? So in this particular problem, we are given our margin of error e as three percent, which is point zero three, and we're also given p hat. That's that point two six because it's 26%. Now the only other thing that we need here in order to find that minimum sample size is our critical z score. Because we have a confidence level of 97%, we can take alpha over two. That's equal to one minus that point nine seven over two, which is 0.015. This then corresponds to a critical z score z alpha over two of 2.17. So now we have everything we need here in order to find that minimum sample size. Remember n that sample size is going to be equal to z alpha over two over e. That whole term gets squared and then this is multiplied by p hat times one minus p hat. Plugging all of our values in here, that's going to be 2.17 over point zero three squaring that term multiplying it by point two six that's p hat then one minus point two six is point seven four now multiplying this out here gives us a value of 1,006.66, but remember we cannot take point six six of a sample, so we need to go ahead and round that up to the next whole number. And the minimum sample size needed here is 1,007 people. Feel free to let us know if you have any questions, and we'll see you in the next one.

8. Sampling Distributions & Confidence Intervals: Proportion

Confidence Intervals for Population Proportion

Statistics for Business

8. Sampling Distributions & Confidence Intervals: Proportion

Confidence Intervals for Population Proportion

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Multiple Choice

You want to make a $97\%$ confidence interval for the population proportion of people between $20-30$ years old who have gotten a speeding ticket in the past $2$ years. A prior study found that $26\%$ of people between $20-30$ years old have received a speeding ticket in the last year. If you want your estimate to be accurate within $3\%$ of the true population proportion, what is the minimum sample size needed?

1007

1006

579

1309

Verified step by step guidance

Identify the key components needed for calculating the sample size for a confidence interval: the desired confidence level, the margin of error, and the estimated proportion from a prior study.

The formula for calculating the sample size for a proportion is:

n = \frac{z^{2} p (1 - p)}{E^{2}}

, where

z

is the z-score corresponding to the confidence level,

p

is the estimated proportion, and

E

is the margin of error.

For a 97% confidence level, find the z-score. This can be done using a standard normal distribution table or calculator. The z-score for 97% confidence is approximately 2.17.

Substitute the values into the formula: