Math  /  Data & Statistics

QuestionIf all other quantities remain the same, how does the indicated change affect the minimum sample size requirement? (a) Increase in the level of confidence (b) Increase in the error tolerance (c) Increase in the population standard deviation (a) How does an increase in the level of confidence affect the minimum sample size requirement? Choose the correct answer below.
An increase in the level of confidence \square the minimum sample size required.

Studdy Solution

STEP 1

1. The formula for minimum sample size n n is given by:
$ n = \left( \frac{Z \cdot \sigma}{E} \right)^2 \]
where Z Z is the Z-score corresponding to the desired confidence level, σ \sigma is the population standard deviation, and E E is the margin of error (error tolerance).
2. We are analyzing how changes in each parameter affect n n while keeping other parameters constant.

STEP 2

1. Analyze the effect of increasing the level of confidence on sample size.
2. Analyze the effect of increasing error tolerance on sample size.
3. Analyze the effect of increasing population standard deviation on sample size.

STEP 3

Understand the relationship between the level of confidence and the Z-score. An increase in the level of confidence results in a higher Z-score.

STEP 4

Since the Z-score increases with higher confidence levels, the term (ZσE)2 \left( \frac{Z \cdot \sigma}{E} \right)^2 increases, leading to a larger sample size requirement.
Conclusion for (a): An increase in the level of confidence \textbf{increases} the minimum sample size required.

STEP 5

Understand the relationship between error tolerance E E and sample size. An increase in E E decreases the fraction ZσE \frac{Z \cdot \sigma}{E} .

STEP 6

Since E E is in the denominator, increasing E E decreases the value of (ZσE)2 \left( \frac{Z \cdot \sigma}{E} \right)^2 , leading to a smaller sample size requirement.
Conclusion for (b): An increase in the error tolerance \textbf{decreases} the minimum sample size required.

STEP 7

Understand the relationship between population standard deviation σ \sigma and sample size. An increase in σ \sigma increases the fraction ZσE \frac{Z \cdot \sigma}{E} .

STEP 8

Since σ \sigma is in the numerator, increasing σ \sigma increases the value of (ZσE)2 \left( \frac{Z \cdot \sigma}{E} \right)^2 , leading to a larger sample size requirement.
Conclusion for (c): An increase in the population standard deviation \textbf{increases} the minimum sample size required.

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