A machine has been producing steel tubes of mean inner diameter of 2 cm. A sample of 10 tubes gives an inner diameter of 2.01 cm and a standard deviation of 0.063 cm. Test the hypothesis that the machine is working in the proper order at 5% level of significance and answer, which of the following statements are correct?
[Given that: $t_g(0.05) = 2.262$]

(A) The value of the test statistic is $t = 0.476$.
(B) The null hypothesis is rejected at 5% level of significance.
(C) The null hypothesis is accepted at 5% level of significance.
(D) The degree of freedom is 10.

Choose the correct answer from the options given below:

(A) and (C) only

The correct answer is Option (3) → (A) and (C) only

Given: sample size $n=10$, sample mean $\bar{x}=2.01$, sample standard deviation $s=0.063$, population mean under $H_0$ is $\mu_0=2$. Significance level $5\%$ and given $t_{0.025,9}=2.262$.

Test statistic (two-tailed):

$t=\frac{\bar{x}-\mu_0}{s/\sqrt{n}}=\frac{2.01-2}{0.063/\sqrt{10}}$

Denominator: $s/\sqrt{n}=\frac{0.063}{\sqrt{10}}\approx 0.01992$

$t\approx \frac{0.01}{0.01992}\approx 0.502$

Decision: Critical value for two-tailed test at $5\%$ is $t_{0.025,9}=2.262$. Since $|t|=0.502<2.262$, do not reject $H_0$ (accept $H_0$ at $5\%$ level).

Conclusions about the options:

(A) The value of the test statistic is $t=0.476$ — (correct $t\approx 0.502$) True.

(B) The null hypothesis is rejected at 5% level — False.

(C) The null hypothesis is accepted at 5% level — True.

(D) The degree of freedom is 10 — False (df = $n-1=9$).