Question

A random sample of U.S. adults taken 4 years ago indicated that 70% were optimistic about the economy. An economist believes that this percentage has increased. (a) What is the variable? Option 1: The percentage of U.S. adults in the population that are optimistic about the economy. Opt. 2: The percentage of U.S. adults in the sample that are optimistic about the economy. Opt. 3: Whether or not the economist believes the percentage of U.S. adults that are optimistic about the economy is more than 70%. Opt. 4: Whether or not a U.S. adult is optimistic about the economy. (b) Which of the following are the correct hypotheses? Opt 1: Ho: p = 0.70, Ha: p ≠ 0.70 Opt 2: Ho: p = 0.70, Ha: p > 0.70 Opt 3: Ho: p = 0.70, Ha: p < 0.70 Opt 4: Ho: p = 0.70, Ha: p > 0.70 Opt 5: Ho: p >= 0.70, Ha: p < 0.70 The economist takes another random sample. Suppose out of 50 people questioned, 41 of them were optimistic about the economy. (c) Find the sample proportion of people who are optimistic about the economy. (3 decimal places) (d) To determine the p-value, which of the following distribution models would we use? Opt 1: Normal if n(0.70) and n(0.30) are both at least 15. Opt 2: Normal if n(0.82) and n(0.18) are both at least 15. Opt 3: T distribution if n(0.70) and n(0.30) are both at least 15. Opt 4: T distribution with 49 degrees of freedom because n is greater than 30. (e) Find the test statistic. (3 decimal places) (f) Find the p-value. (3 decimal places) (g) What is your conclusion at α = 0.05? Opt 1: Do not reject the null hypothesis. There is insufficient evidence that the proportion of people who are optimistic about the economy has increased. Opt 2: Reject the null hypothesis. There is insufficient evidence that the proportion of people who are optimistic about the economy has increased. Opt 3: Do not reject the null hypothesis. There is evidence that the proportion of people who are optimistic about the economy has increased. Opt 4: Reject the null hypothesis. There is sufficient evidence that the proportion of people who are optimistic about the economy has increased.

Name: a random sample of us adults taken 4 years ago indicated that 70 were optimistic about the economy an economist believes that this percent has increased a what is the variable option 1 the p 29197
Uploaded: 2022-03-29T03:01:07-08:00
Duration: 9 min 41 s
Channel: Idabelle Cunningham
Description: a random sample of us adults taken 4 years ago indicated that 70 were optimistic about the economy an economist believes that this percent has increased a what is the variable option 1 the p 29197

          A random sample of U.S. adults taken 4 years ago indicated that 70% were optimistic about the economy. An economist believes that this percentage has increased.

(a) What is the variable?
Option 1: The percentage of U.S. adults in the population that are optimistic about the economy.
Opt. 2: The percentage of U.S. adults in the sample that are optimistic about the economy.
Opt. 3: Whether or not the economist believes the percentage of U.S. adults that are optimistic about the economy is more than 70%.
Opt. 4: Whether or not a U.S. adult is optimistic about the economy.

(b) Which of the following are the correct hypotheses?
Opt 1: Ho: p = 0.70, Ha: p ≠ 0.70
Opt 2: Ho: p = 0.70, Ha: p > 0.70
Opt 3: Ho: p = 0.70, Ha: p < 0.70
Opt 4: Ho: p = 0.70, Ha: p > 0.70
Opt 5: Ho: p >= 0.70, Ha: p < 0.70

The economist takes another random sample. Suppose out of 50 people questioned, 41 of them were optimistic about the economy.

(c) Find the sample proportion of people who are optimistic about the economy. (3 decimal places)

(d) To determine the p-value, which of the following distribution models would we use?
Opt 1: Normal if n(0.70) and n(0.30) are both at least 15.
Opt 2: Normal if n(0.82) and n(0.18) are both at least 15.
Opt 3: T distribution if n(0.70) and n(0.30) are both at least 15.
Opt 4: T distribution with 49 degrees of freedom because n is greater than 30.

(e) Find the test statistic. (3 decimal places)

(f) Find the p-value. (3 decimal places)

(g) What is your conclusion at α = 0.05?
Opt 1: Do not reject the null hypothesis. There is insufficient evidence that the proportion of people who are optimistic about the economy has increased.
Opt 2: Reject the null hypothesis. There is insufficient evidence that the proportion of people who are optimistic about the economy has increased.
Opt 3: Do not reject the null hypothesis. There is evidence that the proportion of people who are optimistic about the economy has increased.
Opt 4: Reject the null hypothesis. There is sufficient evidence that the proportion of people who are optimistic about the economy has increased.

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Allan G. Bluman 9th Edition

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A random sample of U.S. adults taken 4 years ago indicated that 70% were optimistic about the economy. An economist believes that this percentage has increased. (a) What is the variable? Option 1: The percentage of U.S. adults in the population that are optimistic about the economy. Opt. 2: The percentage of U.S. adults in the sample that are optimistic about the economy. Opt. 3: Whether or not the economist believes the percentage of U.S. adults that are optimistic about the economy is more than 70%. Opt. 4: Whether or not a U.S. adult is optimistic about the economy. (b) Which of the following are the correct hypotheses? Opt 1: Ho: p = 0.70, Ha: p ≠ 0.70 Opt 2: Ho: p = 0.70, Ha: p > 0.70 Opt 3: Ho: p = 0.70, Ha: p < 0.70 Opt 4: Ho: p = 0.70, Ha: p > 0.70 Opt 5: Ho: p >= 0.70, Ha: p < 0.70 The economist takes another random sample. Suppose out of 50 people questioned, 41 of them were optimistic about the economy. (c) Find the sample proportion of people who are optimistic about the economy. (3 decimal places) (d) To determine the p-value, which of the following distribution models would we use? Opt 1: Normal if n(0.70) and n(0.30) are both at least 15. Opt 2: Normal if n(0.82) and n(0.18) are both at least 15. Opt 3: T distribution if n(0.70) and n(0.30) are both at least 15. Opt 4: T distribution with 49 degrees of freedom because n is greater than 30. (e) Find the test statistic. (3 decimal places) (f) Find the p-value. (3 decimal places) (g) What is your conclusion at α = 0.05? Opt 1: Do not reject the null hypothesis. There is insufficient evidence that the proportion of people who are optimistic about the economy has increased. Opt 2: Reject the null hypothesis. There is insufficient evidence that the proportion of people who are optimistic about the economy has increased. Opt 3: Do not reject the null hypothesis. There is evidence that the proportion of people who are optimistic about the economy has increased. Opt 4: Reject the null hypothesis. There is sufficient evidence that the proportion of people who are optimistic about the economy has increased.

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Transcript

00:01 A random sample of u .s.

00:02 Adults taken four years ago indicated that 70 % were optimistic about the economy.

00:07 An economist believes that this percent has increased.

00:11 In part a, we're asked for what is the variable? all right, p stands for the proportion of a population.

00:20 So the correct choice here is going to be the percentage.

00:24 Actually, i'm going to even let that equal.

00:25 So p is equal to the percentage of u .s.

00:34 Adult.

00:34 In the population that are optimistic about the economy.

01:03 Okay.

01:06 Part b asks, which of the following are the correct hypotheses? all right.

01:12 So before you've been looking at the choices, our null hypothesis is that the true proportion hasn't changed from four years ago, so it should be 0 .70.

01:23 And then the alternative is what we believe is true, which is what the economist believes, and they believe that this percent has increased.

01:32 So they believe that the true proportion is actually greater than 0 .70.

01:39 Okay, so and out of the choices, that would be option two.

01:46 Okay.

01:50 The economists takes another random sample.

01:53 Suppose that out of, this is part c, the out of 50 people questioned, 41 of them were optimistic about the economy.

02:01 All right, so since this is from a sample, we're going to label that p hat, and that is equal to 41 out of 50, which is 0 .82.

02:21 And part c says, find the sample proportion of people who are optimistic about the economy to 3 decimal places.

02:27 Okay, so 0 .8 to 0 .8 .0.

02:32 Okay.

02:34 Part d says to determine the p value, which of the following distribution, models would we use? okay.

02:48 So since we're in the world of proportions, all right, we want to make sure that the large counts condition is met, and i'm looking at all of the options, and they're looking for greater than or equal to 15, which is interesting because sometimes it might be met if the sample size times the proportion of success and proportion of failure are both greater than or equal to 10, but i'm going to go with what it says here.

03:19 So large counts condition basically says that your sample size times p hat has to be greater than or equal to 15 in this case.

03:30 And your sample size times one minus p hat has to be greater than or equal to 15.

03:38 And we would be able to use the normal distribution if this is true.

03:44 So in this case, case.

03:51 Oh, actually i'm sorry.

03:52 We're not going to be using p hat.

03:54 We're actually going to be using p, uh, since we do have a proportion for the population, which should be, according to our hypothesis test, which should be 0 .7.

04:04 All right.

04:05 So we're going to check this.

04:07 Our sample size was 50.

04:10 So 50 times p, which is 0 .7.

04:17 50 times 0 .7 is 35.

04:22 So that's good.

04:24 And then 50 times 1 minus .7, which is .3, is 15, which is at least 15...

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