A research center claims that more than 80% of U.S. adults thi...

Question

A research center claims that more than 80% of U.S. adults think that mothers should have paid maternity leave. In a random sample of 50 U.S. adults, 82% think that mothers should have paid maternity leave. At α=0.05, is there enough evidence to support the center’s claim?

Accepted Answer

Hello there. Today we're going to 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. A poll suggests that over 75% of city residents support expanding public transportation. In a random sample of 60 residents, 78% expressed support. At alpha equals 0.05, is there sufficient evidence to back the polls claim? Awesome. So it appears for this particular problem we're asked to take a significance level alpha of 0.05, and we're asked to use that to help us determine based on all the other information that is provided to us by the problem itself. Is there sufficient evidence to back the poll's claim? So that's our final answer we're ultimately trying to solve for. And once again, the poll's claim is, is that they're suggesting that over 75% of city residents support expanding public transportation, so that's their claim. So with that in mind, now that we know what we're ultimately trying to solve for, our first step is we need to state our hypotheses. So let's start with the null hypothesis. So the null hypothesis. Which will denote as H0 H subscript 0, states that P is equal to 0.75. Whereas the alternative hypothesis, which will denote as HA, states that P is greater than 0.75, where the 0.75 is the 75% written as a decimal. Awesome. So at this point, it's important to note and recall that we need to be able to recognize that this is a right-tailed test, which I'll call RT for short. So this is a right-tail test. So with that in mind, our second step now is we need to identify our given data, the data that is provided to us by the problem itself. We're told that PAT is equal to 0.78. So once again, PA is equal to 0.78, and this is out of the random sample of 60 residents, this is the 78% that expressed support. And we're told that our sample size N is equal to 60. So now we need to recall a note that we could solve for the claimed population proportion, which we'll call this P0, is equal to 75 divided by 100, which is equal to 0.75. And now for our 3rd step, we need to determine the standard error, which, as we should recall, the standard error which is denoted as SE. is equal to the square root of P0 multiplied by parentheses 1 minus P0 divided by N, which is equal to when we plug in our known variables, 0.75 multiplied by parentheses 1 minus 0.75. Divided by 60. Once again this is in a square root, so the square root of 0.75 multiplied by parentheses 1 minus 0.75 divided by 60, when we plug in this expression into our calculator and we round to 4 decimal places, we will find that it's approximately equal to 0.0559. Fantastic. So now our 4th step is we need to calculate the test statistics. So in order to do that, we need to recall and use the following formula which states that Z is equal to Phat minus P0 divided by SE, which is equal to when we plug in our known variables, 0.78 minus 0.75 divided by 0.0559. Which will be when we plug that expression into our calculator, it'll be approximately equal to 0.5367 when we round to 4 decimal places. Our fifth step is we need to find the critical value for the significance level alpha equals 0.05, noting once again that this is a right tail test. So that will mean that our critical value Z subscript 0.05 will be equal to 1.645. So that will mean that our sixth step now is we need to compare the test statistic to our critical value, which means that since 0.5367 is less than 1.645. We failed to reject the null hypothesis, I'll put an X next to H0 or null hypothesis. Once again, because 0.5367 is less than 1.645, we fail, we fail to reject the null hypothesis. So with that in mind, our seventh and final step is we need to make our final conclusions. Meaning that based on everything we discovered together, there is not sufficient evidence at the significance level of alpha equals 0.05 to support the claim that more than 75% of city residents support expanding public transportation. So that will mean looking at the prom itself. The final answer, without a doubt, will have to be no, there is not sufficient evidence, and that's it. That is our final answer. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

A research center claims that more than 80% of U.S. adults think that mothers should have paid maternity leave. In a random sample of 50 U.S. adults, 82% think that mothers should have paid maternity leave. At α=0.05, is there enough evidence to support the center’s claim?

Key Concepts

Hypothesis Testing

P-Value

Confidence Interval

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