In Constructing a Confidence Interval For $\sigma$ Or $\sigma ^ { 2 }$

question 8

Multiple Choice

In constructing a confidence interval for $\sigma$ or $\sigma ^ { 2 }$ , a table is used to find the critical values $\chi \frac { 2 } { L }$ and $\chi \frac { 2 } { R }$ for values of $n \leq 101$ . For larger values of $n , \chi _ { L } ^ { 2 }$ and $\chi _ { R } ^ { 2 }$ can be approximated by using the following formula: $\chi ^ { 2 } =$ $\frac { 1 } { 2 } [ \pm z / 2 / 2 + \sqrt { 2 k - 1 } ] 2$ where $k$ is the number of degrees of freedom and $z \alpha / 2$ is the critical $z$ -score. Construct the $90 \%$ confidence interval for $\sigma$ using the following sample data: a sample of size $n = 232$ yields a mean weight of $154 \mathrm { lb }$ and a standard deviation of $25.5 \mathrm { lb }$ . Round the confidence interyal limits to the nearest hundredth.

• A) \(23.39 \mathrm { lb } < \sigma < 28.09 \mathrm { lb }\)
• B) \(24.09 \mathrm { lb } < \sigma < 27.15 \mathrm { lb }\)
• C) \(23.71 \mathrm { lb } < \sigma < 27.65 \mathrm { lb }\)
• D) \(23.66 \mathrm { lb } < \sigma < 27.58 \mathrm { lb }\)

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Definitions:

Matched Pairs Sample

A study design where subjects are paired based on certain characteristics or conditions, and then differences between pairs are examined for impact of a treatment or condition.

Frequency Distributions

An arrangement of values that shows the frequency with which each value occurs.

Population Median

The middle value of a dataset when it is ordered from the smallest to the largest, representing the central tendency of the population.

Normal Approximation

Normal approximation is a statistical technique that applies the properties of the normal distribution to approximate the distribution of sample data or statistics.