Math Problem Statement

a) How many subjects are needed to estimate the mean number of books read the previous year within

sixsix

books with

9595​%

​confidence?

This****

95 %95%

confidence level requires

1616

subjects. ​(Round up to the nearest​ subject.)

Part 2

​(b)**** How many subjects are needed to estimate the mean number of books read the previous year within

threethree

books with

9595​%

​confidence?

This

95 %95%

confidence level requires

6464

subjects. ​(Round up to the nearest​ subject.)

Part 3

​(c)**** What effect does doubling the required accuracy have on the sample​ size?

A.

Doubling the required accuracy nearly quadruples the sample size.

Your answer is correct.

B.

Doubling the required accuracy nearly doubles the sample size.

C.

Doubling the required accuracy nearly halves the sample size.

D.

Doubling the required accuracy nearly quarters the sample size.

Part 4

​(d) How many subjects are needed to estimate the mean number of books read the previous year within

sixsix

books with

9999​%

​confidence?

This

9999​%

confidence level requires

2828

subjects. ​(Round up to the nearest​ subject.)

Part 5

Compare this result to part

​(a).

How does increasing the level of confidence in the estimate affect sample​ size? Why is this​ reasonable?

A.

Increasing the level of confidence decreases the sample size required. For a fixed margin of​ error, greater confidence can be achieved with a larger sample size.

B.

Increasing the level of confidence decreases the sample size required. For a fixed margin of​ error, greater confidence can be achieved with a smaller sample size.

C.

Increasing the level of confidence increases the sample size required. For a fixed margin of​ error, greater confidence can be achieved with a larger sample size.

D.

Increasing the level of confidence increases the sample size required. For a fixed margin of​ error, greater confidence can be achieved with a smaller sample size.

Solution

Part 1: Calculations and Explanations

(a) Subjects needed for 95% confidence within 6 books:

The given value states that 16 subjects are required. This result is calculated using the formula for sample size estimation:

n=(zσE)2n = \left(\frac{z \cdot \sigma}{E}\right)^2

Where:

  • zz = critical value for the given confidence level (1.96 for 95%)
  • σ\sigma = population standard deviation (assumed or provided)
  • EE = margin of error (6 books in this case)

(b) Subjects needed for 95% confidence within 3 books:

The result provided indicates 64 subjects are required. Doubling the accuracy (reducing the margin of error from 6 to 3) increases the sample size significantly.

(c) Effect of doubling accuracy on sample size:

When the required accuracy is doubled (margin of error halved), the sample size increases approximately fourfold because the formula for nn includes EE in the denominator squared.

Correct answer: A. Doubling the required accuracy nearly quadruples the sample size.


(d) Subjects needed for 99% confidence within 6 books:

The result given is 28 subjects. The calculation considers the higher critical value (z=2.576z = 2.576) for a 99% confidence level. This leads to a higher sample size compared to part (a).


Part 5: Effect of increasing confidence level on sample size:

Correct answer: C. Increasing the level of confidence increases the sample size required. For a fixed margin of error, greater confidence can be achieved with a larger sample size.

This is reasonable because a higher confidence level means a smaller probability of error, which requires more data to ensure the estimate meets the tighter criteria.


Summary of Correct Answers:

  • (a): 16 subjects
  • (b): 64 subjects
  • (c): A. Doubling the required accuracy nearly quadruples the sample size.
  • (d): 28 subjects
  • Part 5: C. Increasing the level of confidence increases the sample size required.

Follow-Up Questions:

  1. How does the critical value zz change for different confidence levels (e.g., 95%, 99%)?
  2. Why does halving the margin of error lead to a fourfold increase in sample size?
  3. Can you compute the sample size for other confidence levels (e.g., 90%, 98%)?
  4. How does the population standard deviation affect the required sample size?
  5. Why is rounding up the sample size essential in statistical studies?

Tip: Always round up the sample size in calculations to ensure your estimate meets the desired confidence and accuracy levels.

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Math Problem Analysis

Mathematical Concepts

Statistics
Sample Size Estimation
Confidence Intervals
Effect of Margin of Error

Formulas

n = (z * σ / E)^2

Theorems

Central Limit Theorem (used to justify the normal approximation for sample size estimation)

Suitable Grade Level

Grades 11-12 or Introductory College Level