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During the 2004 election year, new polling results were reported daily. In an IBD/TIPP poll of 910 adults, 503 respondents reported that they were optimistic about the national outlook, and President Bush's leadership index jumped 4.7 points to 55.3 ( Investor's Business Daily, January $14,2004$ ).a. What is the sample proportion of respondents who are optimistic about the national outlook?b. A campaign manager wants to claim that this poll indicates that the majority of adults are optimistic about the national outlook. Construct a hypothesis test so that rejection of the null hypothesis will permit the conclusion that the proportion optimistic is greater than 50$\%$ .c. Use the polling data to compute the $p$ -value for the hypothesis test in part (b). Explain to the manager what this $p$ -value means about the level of significance of the results.

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a. $\hat{p}=0.5527$b. $H_{0} : p \leq 0.50, H_{a} : p>0.50$c. $P \approx 0$

Intro Stats / AP Statistics

Chapter 9

Hypothesis Tests

Confidence Intervals

Hypothesis Testing with One Sample

Hypothesis Testing with Two Samples

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in Question 57. We're told that during an election year we had a sample of 910 adults 503 of which were optimistic about the national outlook. And so, in part, they were asked to give the proportion of the sample that are optimistic. So that is simply P bar is equal to you. 503 over 910 and that comes up to 0.55 to 7 for Part B. We're told that someone wishes to use this poll to indicate that a majority of adults are optimistic about the national outlook, and we're has to devise a hypothesis test test this claim. So let's start with alternative hypothesis. So that is P. So the majority of people is more than 50%. So the alternative hypothesis is that P is greater than 0.5, and the null hypothesis is that P is less than or equal to 0.5, and then in part C, we're as to use the polling data to calculate the P value and then to explain in plain language with the P value means. So our test statistic is said because sample proportions follow a normal distribution. So we have that is equal to this. So we get that said it is equal to 3.18 So our test statistic is 3.18 So if we look that up on the normal standard table, the normal standard table in this and this one only goes up to 3.9 So we know that the area to the left of our test statistic is going to be something greater than 0.999 But this is an upper town test. So the P value is the area to the right of the test statistic. So it means that the P value was something close to zero. So it's a lefty. Value is approximately zero. And then to explain what this means, um, the P value is the probability under the assumption of the null hypothesis that we would get a result at least this extreme. So for this particular example, P values the probability of a sample of size 910 returning a proportion at least as high as 0.55 to 7. Since the probability in this case is extremely small. It leads us to believe that the null hypothesis is not true. And so we reject the null hypothesis in favor of the claim that the majority of adults are optimistic about the national outlook.

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