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

question 13

Multiple Choice

In constructing a confidence interval for $\sigma$ or $\sigma ^ { 2 }$ , a table is used to find the critical values $\chi _ { \mathrm { L } } ^ { 2 }$ and $\chi _ { \mathrm { R } } ^ { 2 }$ for values of $n \leq 101$ . For larger values of $n , \chi _ { \mathrm { L } } ^ { 2 }$ and $\chi _ { \mathrm { R } } ^ { 2 }$ can be approximated by using the following formula: $\chi ^ { 2 } = \frac { 1 } { 2 } \pm z _ { \alpha / 2 } + \sqrt { 2 k - 1 } ^ { 2 }$ 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 interval limits to the nearest hundredth.

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

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Definitions:

Induced Consumption

Consumer spending that increases when income increases, and decreases when income decreases, not including autonomous consumption that doesn’t change with income.

Disposable Income

Financial capacity reserved for household spending and savings after navigating through income tax deductions.

Autonomous Consumption

The level of consumption expenditure that occurs when income levels are zero, representing the base level of spending that must occur even in the absence of income.

Public Savings

The portion of national savings held by the government, typically reflected in budget surpluses.