On December 25,2009, an airline passenger was subdued while...

On December $25,2009,$ an airline passenger was subdued while attempting to blow up a Northwest Airlines flight headed for Detroit, Michigan. The passenger had smuggled explosives hidden in his underwear past a metal detector at an airport screening facility. As a result, the Transportation Security Administration (TSA) proposed installing full-body scanners to replace the metal detectors at the nation's largest airports. This proposal resulted in strong objections from privacy advocates, who considered the scanners an invasion of privacy. On January $5-6,2010,$ USA Today conducted a poll of 542 adults to learn what proportion of airline travelers approved of using full-body scanners $(U S A$ Today, January $11,2010$ ). The poll results showed that 455 of the respondents felt that full-body scanners would improve airline security and 423 indicated that they approved of using the devices. a. Conduct a hypothesis test to determine if the results of the poll justify concluding that over 80$\%$ of airline travelers feel that the use of full-body scanners will improve airline security. Use $\alpha=.05$ b. Suppose the TSA will go forward with the installation and mandatory use of full-body scanners if over 75$\%$ of airline travelers approve of using the devices. You have been told to conduct a statistical analysis using the poll results to determine if the TSA should go forward with mandatory use of the full-body scanners. Because this is viewed as a very sensitive decision, use $\alpha=.01 .$ What is your recommendation?

On December $25,2009,$ an airline passenger was subdued while attempting to blow up a Northwest Airlines flight headed for Detroit, Michigan. The passenger had smuggled explosives hidden in his underwear past a metal detector at an airport screening facility. As a result, the Transportation Security Administration (TSA) proposed installing full-body scanners to replace the metal detectors at the nation's largest airports. This proposal resulted in strong objections from privacy advocates, who considered the scanners an invasion of privacy. On January $5-6,2010,$ USA Today conducted a poll of 542 adults to learn what proportion of airline travelers approved of using full-body scanners $(U S A$ Today, January $11,2010$ ). The poll results showed that 455 of the respondents felt that full-body scanners would improve airline security and 423 indicated that they approved of using the devices.
a. Conduct a hypothesis test to determine if the results of the poll justify concluding that over 80$\%$ of airline travelers feel that the use of full-body scanners will improve airline security. Use $\alpha=.05$
b. Suppose the TSA will go forward with the installation and mandatory use of full-body scanners if over 75$\%$ of airline travelers approve of using the devices. You have been told to conduct a statistical analysis using the poll results to determine if the TSA should go forward with mandatory use of the full-body scanners. Because this is viewed as a very sensitive decision, use $\alpha=.01 .$ What is your recommendation?

Key Concepts

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis (H?) represents the default assumption about a population parameter (e.g., that a proportion is equal to a given value), while the alternative hypothesis (H?) represents the claim to be tested (e.g., that the proportion is greater than that value). This framework establishes the basis for statistical decision making.

Z-Test for Population Proportions

The z-test for a population proportion is used when the sample size is sufficiently large for the Central Limit Theorem to apply, allowing the sample proportion to be modeled by a normal distribution. The test statistic is calculated by taking the difference between the sample proportion and the hypothesized population proportion, then dividing by the standard error.

Standard Error and Sample Proportion

The sample proportion, computed from the number of successes divided by the total sample size, provides an estimate of the underlying population proportion. The standard error quantifies the variability of this estimate and is calculated based on the hypothesized proportion and the sample size, playing a crucial role in determining the test statistic.

Significance Level (Alpha)

The significance level, denoted as ?, is the threshold for making decisions in hypothesis testing. It represents the probability of committing a Type I error (rejecting the null hypothesis when it is true) and guides the determination of the critical region for the test.

P-Value and Decision Making

The p-value is the probability of observing the test statistic, or one more extreme, under the null hypothesis. By comparing the p-value to the predetermined significance level (?), one can decide whether to reject or fail to reject the null hypothesis, thereby drawing a statistical conclusion.

Transcript

00:01 This again is a very basic question based on hypothesis testing for proportions.

00:09 So what is happening over here is that usa today has conducted a poll of 542 adults to learn what proportion of airline travelers approved of using full body scanners.

00:22 Okay, so one very important piece of information here is that n is equal to 542, right? this is our sample size.

00:30 Now the poll results showed that 455 of the respondents right 455 respondents felt that the body scanners would improve security they said that this would improve security they said that this would improve security okay and 423 of them 423 of them indicated that they approved of using these devices they approved of using these devices.

01:11 Okay, so let us move on to the questions.

01:14 What does part a say? part a says that we have to conduct the hypothesis test to determine if the results of the poll justify concluding that over 80 % of the airline travelers feel that the use of full body scanners will improve the airline security.

01:34 Okay, so what is going to be the null and the alternative? hypothesis in this case the null and the alternative hypothesis okay so the null hypothesis will be that the proportion is less than or equal to 0 .8 okay and the alternative will be that rp is greater than 0 .8 what are they saying they want to test to determine if the results of the poll justify concluding that over 80 % of airline travelers feel over 80 % so yes the same is going to be our alternative hypothesis p greater than 0 .8 okay now in order to conduct the hypothesis test we are going to use the z statistic and what is the formula for the z statistic formula for z statistic is p bar which is our sample proportion minus p not which happens to be our hypothesized proportion or 0 .8 is what we are going to take in this case upon root over 0 .8 into 20 .2 upon n what is the value of n n is 542 or let me just write the formula away this is the formula right this is what we are going to use so this is going to be upon p not into 1 minus p not upon n this is the formula that we are going to use okay so what is p bar p bar is going to be 455 by 542 p bar is going to be 455 by 542 right this will be 455 by 542 and if i use a calculator for this 45 variable by 542 is 0 .8394 is 0 .8394 is 0 .8394 8394 8394 8394 83948 or let me write this as 8395 okay let's put in the values for the z statistic this is going to be 0 .8395 minus 0 .8 .8 upon p .0 which happens to be 0 .8 multiplied by 1 minus p .0 which is 0 .2 upon n what was n 542 right so this thing actually becomes 0 .0395 upon root over this thing so if i use a calculator for this this is root over 0 .8 into 0 .2 divided by 542 which happens to be 0 .01718 0 .0 .01718.

04:46 So this is 0 .0395 divided by 0 .0718.

04:53 2 .299.

04:56 So let me just write this as 2 .3.

04:59 Right.

05:00 This becomes 2 .3.

05:01 So my z statistic is 2 .3.

05:05 All right.

05:07 So now in order to find the p value, we can use any statistical software that we like or an online tool or maybe just a manual z table.

05:16 So what i'm using over here is a z table.

05:20 My z statistic is 2 .3.

05:23 So we have a very useful visualization here.

05:27 Let me just set my z value to 2 .3.