Ignore others question is A & BHow larger....“A sample size of ... To obtain an 80% confidence interval with a margine of error equal to 3.8”“If the required confidence level were 99.8% would the necessary sample size be larger or smaller” ### Understanding Confidence Intervals and Sample Size Determination#### Scenario:A population has a standard deviation \( \sigma = 17.8 \).#### Part 1 of 2**(a)** How large a sample must be drawn so that an 80% confidence interval for \( \mu \) will have a margin of error equal to 3.8?- **Instructions:** - Round the critical value to no less than three decimal places. - Round the sample size up to the nearest integer.**Answer:**A sample size of [____] is needed to be drawn in order to obtain an 80% confidence interval with a margin of error equal to 3.8.#### Part 2 of 2**(b)** If the required confidence level were 99.8%, would the necessary sample size be larger or smaller?- **Options:** - Larger [ ] - Smaller [ ]**Answer:**Larger [____], because the confidence level is higher.---### Exploring the Concepts:#### Confidence Intervals:A confidence interval provides a range of values which is likely to include an unknown population parameter, derived from the statistics of the observed data. The higher the confidence level, the wider the interval must be to encompass the true population parameter.#### Margin of Error:The margin of error indicates the range within which the true population parameter is expected to lie, with a given level of confidence. For instance, a smaller margin of error requires a larger sample size because it increases the precision of the estimate.#### Sample Size Determination:The sample size required for a specific margin of error can be calculated using the formula for the margin of error in a confidence interval.\[ n = \left(\frac{Z_\alpha \times \sigma}{E}\right)^2 \]Where:- \( n \) is the sample size.- \( Z_\alpha \) is the critical value from the Z-distribution corresponding to the desired confidence level.- \( \sigma \) is the population standard deviation.- \( E \) is the margin of error.#### Example Calculation:To determine the sample size for an 80% confidence interval with \( \sigma = 17.8 \) and \( E = 3.8 \), the Z-value for 80% confidence (which can be found using statistical tables or software) must be applied.Working through the mathematics:\[ Z_{0

Name: Ignore others question is A & BHow larger....“A sample size of ... To
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Description: Ignore others question is A & BHow larger....“A sample size of ... To obtain an 80% confidence interval with a margine of error equal to 3.8”“If the required confidence level were 99.8% would the necessary sample size be larger or smaller” ### Unders