Solve the Problem $\sigma$ Or $\sigma ^ { 2 }$ , a Table Is Used to Find the Critical Values

question 62

Multiple Choice

Solve the problem.
-In constructing a confidence interval for $\sigma$ or $\sigma ^ { 2 }$ , a table is used to find the critical values $\chi _ { L } ^ { 2 }$ and $\chi _ { R } ^ { 2 }$ for values
of $n \leq 101$ . For larger values of $n , \chi _ { L } ^ { 2 }$ and $\chi \frac { 2 } { R }$ can be approximated by using the following formula:
$$\chi ^ { 2 } = \frac { 1 } { 2 } \left[ \pm z _ { \alpha / 2 } + \sqrt { 2 \mathrm { k } - 1 } \right] ^ { 2 }$$
where $\mathrm { k }$ is the number of degrees of freedom and $\mathrm { z } _ { \alpha / 2 }$ is the critical $\mathrm { z }$ score. Estimate the critical values $\chi _ { \mathrm { L } } ^ { 2 }$ and $\chi _ { \mathrm { R } } ^ { 2 }$ for a situation in which you wish to construct a $95 \%$ confidence interval for $\sigma$ and in which the sample size is $\mathrm { n } = 298$ .

• A) \(\chi _ { \mathrm { L } } ^ { 2 } \approx 258.727 , \chi _ { \mathrm { R } } ^ { 2 } \approx 338.979\)
• B) \(\chi _ { L } ^ { 2 } \approx 250.692 , \chi _ { R } ^ { 2 } \approx 346.150\)
• C) \(\chi _ { \mathrm { L } } ^ { 2 } \approx 251.611 , \chi _ { \mathrm { R } } ^ { 2 } \approx 347.230\)
• D) \(x _ { \mathrm { L } } ^ { 2 } \approx 257.795 , \chi _ { R } ^ { 2 } \approx 337.911\)

---

Definitions:

Physiological Needs

Basic essential requirements for human survival, such as food, water, warmth, and rest, fundamental to Maslow's hierarchy of needs.

Esteem Needs

According to Abraham Maslow's hierarchy of needs, these are the fourth level of human needs, involving respect, self-esteem, status, recognition, and strength.

Affiliation Needs

A person's need to feel a sense of involvement and belonging within a social group.

Security Needs

A level in Maslow's hierarchy of needs, focusing on the desire for safety, stability, and protection from physical and emotional harm.