Answer to Question #250969 in Statistics and Probability for napie

Question

A new manufacturing method is supposed to increase the average life span of electronic components, while the variance of the life span is expected to stay the same. Using the previous manufacturing method, the average life span was 110 hours with a variance of 9 hours. The manufacturer measures the life spans of a sample of components manufactured using the new method. The sample of 20 yields a sample mean of 125 for the life spans of the components. Does the data provide sufficient evidence for the claim at the 1% level of significance?

Clearly show how you draw a conclusion when you test this hypothesis, using both methods, i.e.

(a) critical value approach, and (b) p-value approach.

Answer 1

$\mu=110 \\ s^2 = 9 \\ s=3\\ n=20 \\ \bar{x} = 125 \\ α=0.01 \\ H_0: \mu=110 \\ H_1: \mu>110$

(a) Since, n<30 and $\sigma$ is unknown then we are using T-test statistic

$t = \frac{\bar{x} - \mu}{s / \sqrt{n}} \\ t = \frac{125-110}{3/ \sqrt{20}} = 22.36 \\ df=n-1=20-1=19 \\ t_{0.01,19} = 2.539 \\ t> t_{0.01,19}$

We reject H₀.

(b) P-value approach.

$t=22.6 \\ df=19$

Using Excel formula

The EXCEL formula to find the p-value for a one-tailed t-test and df=19 is

=tdist(22.6, 19, 1)

p-value $= 2.07 \times 10^{-15}$

p ≈ 0<α=0.01

Reject H₀.

Therefore, there is not enough evidence to conclude that a new manufacturing method is supposed to increase the average life span of electronic components, while the variance of the life span is expected to stay the same at 0.01 significance level.