Question

QUESTION 1

1. A recent random survey of 1,000 Americans asked them what they would do with an unexpected tax return. 470 of them indicated they would use it to pay down debt.

   
   Calculate a 90% confidence interval for the proportion of all people who would use an unexpected tax refund to pay debt.

   
   1. Calculate p-tilde, the initial estimate for the proportion of people who would pay down debt. Use five decimals for accuracy.

   a. 470/1000=0.47000 b.(471)/(1002)=0.47006 c(472)/(1004)=0.47012

2. The formula to use for this problem is a. xbar +/- s*[t/square root(n)] b. xbar +/- t*[s/square root(n)] c. xbar +/- Z*[sigma/square root(n)] d. p-tilde +/- Z * square root of[p-tilde*(1- p-tilde)/sqrt(n+4)]

3. What is the correct table to use for this problem? a. Table A because sigma is assumed to be known b. Table F because it is a chi-square problem c.Table D and use the bottom row labeled Z

4. The facts of this problem are:    a. p-tilde=90%;n=1000;confidence=0.47012 b.p-tilde=0.47012;n=1000;confidence=99% c.p-tilde=0.47012;n=1000;confidence=90%

5. The z value to use from the table is a. 2.046 b. 2.064 c.1.960 d.1.645

6. The margin of error for this problem is a. 1.96000 b.0.02596 c.0.18120 d.0.01578

7. The confidence interval for the problem is [remember a confidence interval has the form (lower,upper)] a.(44%,50%) b.(50%,44%) c.(-1.96,1.96) d.(2.6%,47%)

8. A recent journal article stated that the percent of people who would use an unexpected tax return to pay down debt could be as high as 50%. Based on the confidence interval you calculated...    a.This claim is OK because the upper bound of the confidence interval is 50% b.This claim is NOT OK because the upper bound of the confidence interval is 50%

Answer 1

1. Given:

Sample size = n = 1000 Let X = Event that an American would use the unexpected tax return to pay down debt. The event of the required event is considered success.Let p be the probability of Succes. This parameter can be obtained as follows:

Hence, the initial estimate for the proportion of people who would pay down debt is:

a. 470/1000=0.47000

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A 100(1-)% CI for proportion is given by the formula:

Hence, a 90% CI for proportion is given by the formula:

d. p-tilde +/- Z * square root of [p-tilde*(1- p-tilde)/sqrt(n)]

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Given: , by Normal approximation,

. Hence, the appropriate table here, would be a Standard Normal table by computing standard normal variate Z:

c.Table D and use the bottom row labeled Z

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The facts of this problem are:

, confidence=90%

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We must look for

i.e Look for the probability 0.95 in the standard normal table and then find thevalue for which this probability corresponds to:

Hence, the z value to use from the table is d.1.645

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The Margin of Error required to compute the confidence interval is given by the formula:

= 0.02596

The margin of error for this problem is b.0.02596

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A 90% CI for proportion is given by the formula:

=

= (0.44,0.50)

The confidence interval for the problem is a.(44%,50%)

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A recent journal article stated that the percent of people who would use an unexpected tax return to pay down debt could be as high as 50%. Based on the confidence interval calculated above a.This claim is OK because the upper bound of the confidence interval is 50%