In February 2008, the Gallup organization surveyed 1,034 adults between 30 and 64 years of age...

Question

Question

In February 2008, the Gallup organization surveyed 1,034 adults between 30 and 64 years of age and found that 548 were worried that they will outlive their money after they retire. Does the sample evidence suggest that a majority of 30- to 64-year-olds in the United States are worried they will outlive their money? Use the a 0.05 level 2. of significance: (a) Regarding a population proportion, what condition must be provided? (b) Determine the alternative hypothesis if the null hypothesis is p (c) Show the critical value approach. (d) Show the P-value approach. 0.5. (e) Decide whether you reject the null hypothesis or not. Then state the conclusion.

math Statistics-And-Probability

Add a comment Improve this question Transcribed image text

Answer 1

Answer #1

Solution:

Given:

n = sample size = Number of adults surveyed between 30 and 64 years of age = 1034

x = Number of adults worried that they will outlive their money after they retire = 548

We have to test if their is sample evidence to suggest that a majority of 30-to 64-years-old's in the United Status are worried they will outlive their money.

Level of significance = $\alpha = 0.05$

Part a)

Regarding population proportion, following condition must me provided:

$n \times p \geq 10$

and $n \times (1-p) \geq 10$

where p = population proportion = 0.5

$n \times p = 1034 \times 0.5 = 517 > 10$

and

$n \times (1-p) = 1034 \times (1-0.5) = 1034 \times 0.5 = 517 > 10$

Thus both the conditions are satisfied.

Thus sampling distribution of sample proportions is approximately Normal with mean of sample proportions is:

$\mu_{\hat{p}} = p$

$\mu_{\hat{p}} = 0.5$

and standard deviation of sample proportions is:

$\sigma_{\hat{p}}= \sqrt{\frac{p \times (1-p)}{n}}$

$\sigma_{\hat{p}}= \sqrt{\frac{0.5 \times (1-0.5)}{1034}}$

$\sigma_{\hat{p}}= \sqrt{\frac{0.5 \times 0.5 }{1034}}$

$\sigma_{\hat{p}}= \sqrt{ 0.000241779 }$

$\sigma_{\hat{p}}= 0.015549$

Part b)

Null hypothesis is : p = 0.5

Thus alternative hypothesis is: p > 0.5, since we have to test majority of 30-to 64-years-old's in the United Status are worried they will outlive their money, that is: we have to test population proportion is more than 0.5.

Thus

H0: p= 0.5 Vs H1: p > 0.5

Part c) Critical value approach:

We need to find z test statistic value and z critical value.

z test statistic value:

$z= \frac{\hat{p}-\mu_{\hat{p}}}{ \sigma_{\hat{p}} }$

where

$\hat{p} = \frac{x}{n} = \frac{548}{1034}=0.53$

thus

$z= \frac{\hat{p}-\mu_{\hat{p}}}{ \sigma_{\hat{p}} }$

$z= \frac{0.53 - 0.5}{ 0.015549 }$

$z= \frac{0.03}{ 0.015549 }$

z= 1.93

and

z critical value:

Since this is right tailed test, find area = 1 - 0.05 = 0.95

Look in z table for Area = 0.9500 or its closest area and find corresponding z value.

Area 0.9500 is in between 0.9495 and 0.9505 and both the area are at same distance from 0.9500

Thus we look for both area and find both z values

Thus Area 0.9495 corresponds to 1.64 and 0.9505 corresponds to 1.65

Thus average of both z values is : ( 1.64+1.65) / 2 = 1.645

Thus Z_critical = 1.645

Part d) Show the P-value approach:

P-value = P( Z > z test statistic )

P-value = P( Z > 1.93 )

P-value = 1 - P( Z < 1.93 )

Look in z table for z = 1.9 and 0.03 and find corresponding area.

P( Z < 1.93 ) = 0.9732

Thus

P-value = 1 - P( Z < 1.93 )

P-value = 1 - 0.9732

P-value = 0.0268

Part e) Decision and Conclusion:

Decision using Critical value:

Since this is right tailed test, rejection region would be:

Reject null hypothesis ,if z test statistic value > z critical value = 1.645 , otherwise we fail to reject H0.

Since z test statistic value = 1.93 > z critical value = 1.645 , we reject null hypothesis.

Decision using P-value:

Reject H0, if P-value < 0.05 level of significance, otherwise we fail to reject H0.

Since P-value = 0.0268 < 0.05 level of significance, we reject null hypothesis.

Conclusion:

Since we have rejected null hypothesis, we have enough evidence to suggest that a majority of 30-to 64-years-old's in the United Status are worried they will outlive their money.

Add a comment

Answer 2

Similar Homework Help Questions

In February 2008, the Gallup organization surveyed 1,034 adults between 30 and 64 years of age...

In February 2008, the Gallup organization surveyed 1,034 adults between 30 and 64 years of age and found that 548 were worried that they will outlive their money after they retire. Does the sample evidence suggest that a majority of 30- to 64-year-olds in the United States are worried they will outlive their money? Use the a 0.05 level 2. of significance: (a) Regarding a population proportion, what condition must be provided? (b) Determine the alternative hypothesis if the null...
1. In June 2013, the Gallup organization had surveyed 1134 American adults and found that 360...

1. In June 2013, the Gallup organization had surveyed 1134 American adults and found that 360 had confidence in public schools. A year later, in June 2014, Gallup surveyed 1134 American adults and found that 298 had confidence in the public schools. A newspaper reporter saw the data and wrote the headline "Confidence in Schools Deteriorates.'" Test the hypothesis that the of proportion of Americans who have confidence in the public school systems is significantly less in 2014 than in...
In 1944, an organization surveyed 1100 adults and asked, "Are you a total abstainer from, or...

In 1944, an organization surveyed 1100 adults and asked, "Are you a total abstainer from, or do you on occasion consume, alcoholic beverages?" Of the 1100 adults surveyed, 440 indicated that they were total abstainers. In a recent survey, the same question was asked of 1100 adults and 407 indicated that they were total abstainers. Complete parts (a) and (b) below. Find the test statistic for this hypothesis test. (Round to two decimal places as needed.) Determine the P-value for...

In 1940, an organization surveyed 1100 adults and asked, "Are you a total abstainer from, or...

In 1940, an organization surveyed 1100 adults and asked, "Are you a total abstainer from, or do you on occasion consume, alcoholic beverages? Of the 1100 adults surveyed, 363 indicated that they were total abstainers. In a recent survey, the same question was asked of 1100 adults and 330 indicated that they were total abstainers. Complete parts (a) and (b) below. (a) Determine the sample proportion for each sample. respectively. The proportions of the adults who took the 1940 survey...
In 1948, an organization surveyed 1100 adults and asked, "Are you a total abstainer from, or...

In 1948, an organization surveyed 1100 adults and asked, "Are you a total abstainer from, or do you on occasion consume, alcoholic beverages?" Of the 1100 adults surveyed, 385 indicated that they were total abstainers. In a recent survey, the same question was asked of 1100 adults and 341 indicated that they were total abstainers. Complete parts (a) and (b) below. (a) Determine the sample proportion for each sample. and respectively. The proportions of the adults who took the 1942...
In 1941, an organization surveyed 1100 adults and asked, "Are you a total abstainer from, or...

In 1941, an organization surveyed 1100 adults and asked, "Are you a total abstainer from, or do you on occasion consume, alcoholic beverages?" Of the 1100 adults surveyed, 374 indicated that they were total abstainers. In a recent survey, the same question was asked of 1100 adults and 319 indicated that they were total abstainers. Complete parts (a) and (b) below. (a) Determine the sample proportion for each sample. The proportions of the adults who took the 1941 survey and...

1, In 1943, an organization surveyed 1100 adults and asked, "Are you a total abstainer from,...

1, In 1943, an organization surveyed 1100 adults and asked, "Are you a total abstainer from, or do you on occasion consume, alcoholic beverages?" Of the 1100 adults surveyed, 385 indicated that they were total abstainers. In a recent survey, the same question was asked of 1100 adults and 363 indicated that they were total abstainers. Complete parts (a) and (b) below. (a) Determine the sample proportion for each sample. The proportions of the adults who took the 1943 survey...
In 1941, an organization surveyed 1100 adults and asked, "Are you a total abstainer from, or do y...

In 1941, an organization surveyed 1100 adults and asked, "Are you a total abstainer from, or do you on occasion consume, alcoholic beverages?" Of the 1100 adults surveyed, 374 indicated that they were total abstainers. In a recent survey, the same question was asked of 1100 adults and 319 indicated that they were total abstainers. Complete parts (a) and (b) below. (a) Determine the sample proportion for each sample. The proportions of the adults who took the 1941 survey and...
A May 8, 2008, report on National Public Radio (www.npr.org) noted that the average age of...

A May 8, 2008, report on National Public Radio (www.npr.org) noted that the average age of first- time mothers in the United States is slightly higher than 25 years. Suppose that a recently taken random sample of 57 first-time mothers from Missouri produced an average age of 23.90 years and that the population standard deviation is known to be 4.80 years. a. Find the p-value for the test of hypothesis with the alternative hypothesis that the current mean age of...

Gallup recently conducted a survey about the proportion of Americans who use e-cigarettes (vape). The survey data was co...

Gallup recently conducted a survey about the proportion of Americans who use e-cigarettes (vape). The survey data was collected via random-digit-dial telephone interviews conducted July 1-12, 2019 with a random sample of 1525 adults living in all 50 U.S. states with a minimum quota of 70% cellphone respondents and 30% landline respondents, with additional minimum quotas by time zone within region. The following table summarizes their results for men and women: Men Women Total Vape 92 31 123 Do Not...

In February 2008, the Gallup organization surveyed 1,034 adults between 30 and 64 years of age...

Homework Answers

Add Answer to:
In February 2008, the Gallup organization surveyed 1,034 adults between 30 and 64 years of age...

Post as a guest

Earn Coins