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: \(t_9(0.05) = 2.262\)]
(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:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A) and (C) only
Let \(H_0: \mu = 2\) (the machine works properly) against \(H_1: \mu \ne 2\). Here \(n = 10\), \(\bar{x} = 2.01\) and \(s = 0.063\).
The standard deviation 0.063 is that of the sample itself (computed with divisor n). With this s, the one-sample t statistic is
\( \displaystyle t = \frac{\bar{x} - \mu}{s/\sqrt{n-1}} \)
\( \displaystyle = \frac{2.01 - 2}{0.063/\sqrt{9}} \)
\( \displaystyle = \frac{0.01 \times 3}{0.063} \)
\( \displaystyle \approx 0.476 \)
The degrees of freedom are \(n - 1 = 9\), and the table value is \(t_9(0.05) = 2.262\).
\( \displaystyle |t| = 0.476 \lt 2.262 \)
The difference is not significant.
So \(t \approx 0.476\), the degrees of freedom are 9, and \(H_0\) is accepted at 5% level of significance.
Checking the statements
(A) The value of the test statistic is \(t = 0.476\). — Correct. \(t = \frac{0.01 \times 3}{0.063} \approx 0.476\).
(B) The null hypothesis is rejected at 5% level of significance. — Incorrect. \(|t| = 0.476 \lt 2.262\), so \(H_0\) is not rejected.
(C) The null hypothesis is accepted at 5% level of significance. — Correct. \(|t| \lt 2.262\); the machine may be taken to be working properly.
(D) The degree of freedom is 10. — Incorrect. The degrees of freedom are \(n - 1 = 9\).