Critical Thinking. In Exercises 17–28, use the data and confid...

Question

Critical Thinking. In Exercises 17–28, use the data and confidence level to construct a confidence interval estimate of p, then address the given question.

Births A random sample of 860 births in New York State included 426 boys. Construct a 95% confidence interval estimate of the proportion of boys in all births. It is believed that among all births, the proportion of boys is 0.512. Do these sample results provide strong evidence against that belief?

Accepted Answer

Below there, today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. In a survey of 950 households, 418 reported having at least one pet. Construct a 95% confidence interval estimate of the proportion of all households with at least one pet. It is believed that the true proportion is 0.45. Do these results provide strong evidence against that belief? Awesome. So it appears for this particular problem we're asked to solve for two separate answers. Our first answer is we're asked to construct or create or determine a 95% confidence interval to help us estimate the proportion of all households having at least one pet. That's our first answer. Our second answer is we're asked to state whether or not the results of getting a true proportion of 0.45 provides strong evidence against this belief. So do this results of having a true proportion of 0.45 do these results provide strong evidence against that belief? So we're trying to figure out a statement. And that is our second answer we're ultimately trying to solve for. So with that in mind, our first step that we need to take in order to solve this particular problem is we need to write down all of our known variables, all the variables that are given to us by the problem itself. So we're told that the sample size, which we'll call N, and the sample size N is equal to 950. We're also told the number. Of individuals that whole that have a or rather the number of households that have at least one pet. And that value will call X and X is equal to 418. And once again, these are the households that report having at least one pet. And our sample proportion, which we'll call this value Phat, and Phat is equal to X divided by N, which is going to be equal to when we plug in our known variables, 418 divided by 950, which will give us a value of 0.44 when we round to two decimal places. And we're also told the confidence level, which I'll denote as CL, and the confidence level or the con, actually we could instead of calling it CL, we call it CI confidence interval. is equal to 95%. And we need to note that for a 95% confidence interval, the critical value, which will denote this value as Z subscript A divided by 2, and Z subscript A divided by 2 is going to be equal to 1.96. Fantastic. So now that we've defined all of our variables, our first step to solve this particular problem is we need to determine our best point estimate of Phat. So Phat is going to be equal to 0. 44, so that's a good point estimate value that we could use. Our second step is we need to determine the margin of error, so we need to recall and use the margin of error equation, which will state that E, the margin of error, is equal to the subscript alpha divided by A multiplied by the square root of Phat multiplied by parentheses 1 minus Phat. All divided by N. So when we plug in our known variables, that would mean that E, our margin of error, is equal to 1.96 multiplied by the square root of 0.44 multiplied by 1 minus 0.44, all divided by 950. And when we plug that into our calculator and we round to four decial places, we'll get 0.03. 16. Fantastic. So now, Our third step is we need to construct the 95% confidence interval. So we need to note that the confidence interval, which I put up two dots to separate the CNI, but we just call it CI, the confidence interval is going to be equal to Phat plus or minus E or margin of error, which is going to be equal to 0.4. 4 plus or minus 0.0316. So that would mean that CI or confidence interval, will be equal to parentheses 0.4084, 0.4716, which will be approximately equal to 0.41.0. 47. Fantastic in parentheses. And since the believed proportion of 0.45 is within our interval, the sample results do not provide strong evidence against that belief, so that will mean that our final answer or answers I should say, without a doubt, have to be the following. Which as a quick summary. To quickly note here, so our first, our first answer is parentheses 0.41.0.47. So that is our confidence interval. And our second answer is no because 0.45 is within the confidence interval. And that's it. We've officially solved for our final answers. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Key Concepts

Confidence Interval

Proportion

Hypothesis Testing

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