In: Statistics and Probability

A. You want to obtain a sample to estimate a population proportion. Based on previous evidence, you believe the population proportion is approximately p∗=28%p∗=28%. You would like to be 99.5% confident that your esimate is within 2.5% of the true population proportion. How large of a sample size is required?*n* =___

B. Giving a test to a group of students, the grades and gender are summarized below

A | B | C | Total | |

Male | 17 | 15 | 16 | 48 |

Female | 7 | 6 | 9 | 22 |

Total | 24 | 21 | 25 | 70 |

Let ππ represent the percentage of all female students who would receive a grade of B on this test. Use a 95% confidence interval to estimate ππ to three decimal places.

Enter your answer as a tri-linear inequality using decimals (not percents). ___

C. Assume that a sample is used to estimate a population proportion *p*. Find the 98% confidence interval for a sample of size 111 with 49 successes. Enter your answer as an **open-interval** (*i.e.*, parentheses) using decimals (not percents) accurate to three decimal places. 98% C.I. =___

A)

sample proportion , p̂ =
0.28

sampling error , E = 0.025

Confidence Level , CL= 0.995

alpha = 1-CL = 0.005

Z value = Zα/2 = 2.807 [excel
formula =normsinv(α/2)]

Sample Size,n = (Z / E)² * p̂ * (1-p̂) = (
2.807 / 0.025 ) ² *
0.28 * ( 1 - 0.28 ) =
2541.59

so,Sample Size required=
2542

B)

Level of Significance, α =
0.05

Number of Items of Interest, x =
6

Sample Size, n = 70

Sample Proportion , p̂ = x/n =
0.086

z -value = "Zα/2 =

" 1.9600 [excel formula
=NORMSINV(α/2)]

Standard Error , SE = √[p̂(1-p̂)/n] =
0.0335

margin of error , E = Z*SE = 0.0656

Confidence Interval

Interval Lower Limit , = p̂ - E =
0.0201

Interval Upper Limit , = p̂ + E =
0.1513

(0.020

c)

Level of Significance, α =
0.02

Number of Items of Interest, x =
49

Sample Size, n = 111

Sample Proportion , p̂ = x/n =
0.441

z -value = "Zα/2 =

" 2.3263 [excel formula =NORMSINV(α/2)]

Standard Error , SE = √[p̂(1-p̂)/n] =
0.0471

margin of error , E = Z*SE =
0.1096

Confidence Interval

Interval Lower Limit , = p̂ - E =
0.3318

Interval Upper Limit , = p̂ + E =
0.5511

(0.332

You want to obtain a sample to estimate a population proportion.
Based on previous evidence, you believe the population proportion
is approximately p = 0.37. You would like to be 98% confident that
your estimate is within 3% of the true population proportion. How
large of a sample size is required?

You want to obtain a sample to estimate a population proportion.
Based on previous evidence, you believe the population proportion
is approximately p ? = 80 % . You would like to be 99.9% confident
that your esimate is within 2.5% of the true population proportion.
How large of a sample size is required?

You want to obtain a sample to estimate a population proportion.
Based on previous evidence, you believe the population proportion
is approximately p = 0.14. You would like to be 98% confident that
your esimate is within 3% of the true population proportion. How
large of a sample size is required? n=____

You want to obtain a sample to estimate a population proportion.
Based on previous evidence, you believe the population proportion
is approximately 61%. You would like to be 90% confident that your
estimate is within 1% of the true population proportion. How large
of a sample size is required? n=_____

You want to obtain a sample to estimate a population proportion.
Based on previous evidence, you believe the population proportion
is approximately p∗=84%p∗=84%. You would like to be 95% confident
that your esimate is within 3% of the true population proportion.
How large of a sample size is required?

You want to obtain a sample to estimate a population mean. Based
on previous evidence, you believe the population standard deviation
is approximately σ = 55.8 σ = 55.8 . You would like to be 95%
confident that your estimate is within 10 of the true population
mean. How large of a sample size is required?

You want to obtain a sample to estimate a population proportion.
At this point in time, you have no reasonable preliminary
estimation for the population proportion. You would like to be 90%
confident that you estimate is within 3.5% of the true population
proportion. How large of a sample size is required?

You want to obtain a sample to estimate a population proportion.
At this point in time, you have no reasonable estimate for the
population proportion. You would like to be 99% confident that you
esimate is within 5% of the true population proportion. How large
of a sample size is required?
n = ?

You want to obtain a sample to estimate a population proportion.
At this point in time, you have no reasonable preliminary
estimation for the population proportion. You would like to be 90%
confident that you estimate is within 4% of the true population
proportion. How large of a sample size is required? n= ____

You want to obtain a sample to estimate a population proportion.
At this point in time, you have no reasonable estimate for the
population proportion. You would like to be 98% confident that you
esimate is within 0.1% of the true population proportion. How large
of a sample size is required?

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