A company 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 . If \( t_9 \) (0.05) = 2.262, then which of the following is NOT correct?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → The difference between sample mean and population mean is significant.
Null hypothesis \( H_0:\ \mu=2 \) (no significant difference). Here n = 10, \( \bar{x}=2.01 \) and s = 0.063, with n − 1 = 9 degrees of freedom.
With s the standard deviation of the sample, the one-sample t statistic is
\( \displaystyle t=\frac{\bar{x}-\mu}{s/\sqrt{n-1}} \)
\( \displaystyle =\frac{2.01-2}{0.063/\sqrt{9}} \)
\( \displaystyle =\frac{0.01\times 3}{0.063} \)
\( \displaystyle \approx 0.476 \)
Since |t| = 0.476 < 2.262, the null hypothesis is accepted at the 5% level.
Option (1) There is no significant difference between sample mean and population mean at 5 % level of significance. — Correct. |t| = 0.476 is below the table value 2.262.
Option (2) Null hypothesis is accepted at 5% level of significance. — Correct. |t| = 0.476 < 2.262.
Option (3) The value of t-test statistic is 0.476. — Correct. t = 0.03 ÷ 0.063 ≈ 0.476.
Option (4) The difference between sample mean and population mean is significant. — Incorrect. H0 is accepted, so the difference is not significant.
The question asks for the statement that is NOT correct, so the answer is Option (4), the only statement that is not true.