Question

The piston diameter of a certain hand pump is 0.6 inch. the manager determines that the diameters are normally​ distributed, with a mean of 0.6 inch and a standard deviation of 0.003 inch. after recalibrating the production​ machine, the manager randomly selects 26 pistons and determines that the standard deviation is 0.0026 inch. is there significant evidence for the manager to conclude that the standard deviation has decreased at the alpha equals 0.10 level of​ significance?

Answer 1

The given data:

$n=26 \\\mu=0.6 \\ \sigma = 0.003 \\s=0.0026\\\alpha=0.10$

Null Hypothesis: $H_0: \sigma =0.003$

Alternative Hypothesis: $H_1: \sigma<0.003$

Level of significance: $\alpha=0.10$

Test Statistics:

Using chi-square test,

$\chi ^2=\frac{ns^2}{\sigma^2}$

$\Rightarrow \chi ^2=\frac{26*0.0026^2}{0.003^2}=19.528$

Number of degrees of freedom$= n-1=26-1=25.$

Table value of $\chi^2$ for 25 degrees of freedom at 10% level = 34.382.

Conclusion:

There is no significant evidence for the manager to conclude that the standard deviation has decreased.