Calculate the confidence interval to estimate the population proportion for each of the followin

step 1 So we're calculating confidence intervals for proportions. In our first one, we know we might ha

Calculate the confidence interval to estimate the population...

Key Concepts

Standard Error

The standard error is a measure of the variability of the sample proportion as an estimate of the population proportion. It is calculated using the formula sqrt[p*(1-p)/n], where p is the sample proportion and n is the sample size, and it quantifies the precision of the estimated confidence interval.

Margin of Error

The margin of error represents the maximum expected difference between the true population proportion and the estimated sample proportion. It is obtained by multiplying the critical value by the standard error, and it defines the width of the confidence interval around the sample estimate.

Critical Value

The critical value is a point on the distribution, usually taken from the standard normal (z) distribution for large samples, that corresponds to the desired level of confidence. It defines the number of standard errors that extend from the sample estimate to form the bounds of the confidence interval.

Population Proportion

The population proportion is a measure that indicates the fraction of the entire population which exhibits a particular attribute or outcome. When estimating this parameter, the sample proportion serves as an estimate, and statistical techniques are used to infer the true value within a calculated interval.

Confidence Interval

A confidence interval is a range of values, derived from sample data, that is likely to contain the true population parameter with a specified level of certainty. It reflects the uncertainty inherent in estimating a population characteristic, such as a proportion, from a limited sample.

Transcript

00:01 So we're calculating confidence intervals for proportions.

00:05 In our first one, we know we might have a 98 % confidence level.

00:10 Our sample size is 450, and our p hat is 0 .1.

00:16 And so our alpha level is 0 .02.

00:19 So our z for 0 .01, 1 % in the upper tail, is the 2 .326.

00:30 So we're going to have our proportion.

00:34 Whoops, i want to write the actual proportion.

00:38 Our proportion is 0 .1, plus or minus, our 2 .326 times the square root of, and then our proportion, our sample proportion, times its complement, divided by the sample size.

00:56 And so i'm going to plug this in just like this.

01:00 I'm not even going to find what the margin of error is.

01:02 I'm just going to plug it in right away.

01:04 0 .1 minus 2 .326, and then times square root of, and we have 0 .1 times 0 .9 divided by 450.

01:15 I never used any parentheses.

01:17 I just hit enter.

01:18 And my lowest is 0 .0671.

01:21 So about 6 .71 percent.

01:25 Our estimate, point estimate is 10%.

01:29 Two and now we'll change that into an addition sign to find the upper proportion and silly me i actually forgot to change it to an addition sign and just got the same answer twice so 0 .132 and it would round to 9 so 13.

01:51 About 3%.

01:52 So we are 98 % confident that the actual proportion is in here.

01:58 Now in part b, given the information just a little bit different way, actually not, just different numbers.

02:05 We have 95 % confidence interval, our sample size is 240, and our p hat is now smaller, 0 .01, and the alpha level then is 0 .05, and the z value with 0 .025 in the upper tail, or the 95 % number is at 1 .960.

02:24 And we have .01 plus or minus that 1 .96 times the square root of .01 times .99 and then divided by the sample size of 240.

02:39 And again, i'm pointing out specific very .01 minus 1 .96 and then times the square root .01 times .90 times .99 divided by .90 divided by 240...

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