Question

# Assume that we want to construct a confidence interval. Do one of the following, as appropriate: a) find the critical value t_{\alpha/2}. b) find the critical value z_{\alpha/2}, or c) state that neither the normal distribution nor the t distribution applies. Here are summary statistics for randomly selected weights of newborn girls: n=249, \overline{x}=31.5 hg, s=6.3 hg.

Confidence intervals
Assume that we want to construct a confidence interval. Do one of the following, as appropriate: a) find the critical value $$\displaystyle{t}_{{\frac{\alpha}{{2}}}}$$. b) find the critical value $$\displaystyle{z}_{{\frac{\alpha}{{2}}}}$$, or c) state that neither the normal distribution nor the t distribution applies.
Here are summary statistics for randomly selected weights of newborn girls: $$\displaystyle{n}={249},\overline{{{x}}}={31.5}{h}{g},{s}={6.3}{h}{g}$$. The confidence level is $$\displaystyle{95}\%$$.
A. $$\displaystyle{t}_{{\frac{\alpha}{{2}}}}=?$$
B. $$\displaystyle{z}_{{\frac{\alpha}{{2}}}}=?$$
C. Neither the normal distribution nor the t distribution applies.

2021-08-01

Step 1
Introduction:
It is known that t test is used to test the hypothesis for a population mean when the sample size is not large enough $$\displaystyle{\left({<}{30}\right)}$$ and/or the population variance is not known.
In this case the sample size is large enough, however the population variance is not mentioned. The sample standard deviation is used to estimate the population standard deviation.
Thus, it would be appropriate to use the t test with degrees of freedom of $$\displaystyle{n}-{1}$$.
Step 2
Calculation:
Here, the level of significance is $$\displaystyle\alpha={0.05}$$.
The critical value for the two-tailed test is: $$\displaystyle{P}_{{{H}{0}{\left({\mid}{t}_{{{n}-{1}}}\right\rbrace}{\mid}\geq{t}_{{\alpha,{n}-{1}}}}}={0.05}$$, where $$\displaystyle{t}_{{\alpha,{n}-{1}}}$$ is the critical value. The critical value is such that $$\displaystyle{P}_{{{H}{0}{\left({\mid}{t}_{{{n}-{1}}}\right\rbrace}\geq{t}_{{\frac{\alpha}{{2}},{n}-{1}}}}}={P}_{{{H}{0}}}{\left({t}_{{{n}-{1}}}\leq-{t}_{{\frac{\alpha}{{2}},{n}-{1}}}\right)}={0.025}$$
The degrees of freedom is $$\displaystyle{248}{\left(={249}-{1}\right)}$$
To obtain the exact critical value we have used Excel.
Using Excel formulae: =T.INV.2T(0.05,248), the critical value is 1.97.
Thus, the correct option is A and $$\displaystyle{t}_{{{\frac{{\alpha}}{{{2}}}}}}={1.98}$$.