Question

Lightbulbs of a certain type are advertised as having an average lifetime of 750 hours. The price of these bulbs is very favorable, so a potential customer has decided to go ahead with a purchase arrangement unless it can be conclusively demonstrated that the true average lifetime is smaller than what is advertised. A random sample of 40 bulbs was selected, the lifetime of each bulb determined, and the appropriate hypotheses were tested using MINITAB, resulting in the accompanying output.

Variable	N	Mean	StDev	SEMean	Z	P-Value
lifetime	40	738.44	38.13	6.03	−1.92	0.028

What conclusion would be appropriate for a significance level of 0.05?

Do not reject the null hypothesis. There is sufficient evidence to conclude that the lifetime of a bulb is less than 750 hours.Reject the null hypothesis. There is not sufficient evidence to conclude that the lifetime of a bulb is less than 750 hours.    Do not reject the null hypothesis. There is not sufficient evidence to conclude that the lifetime of a bulb is less than 750 hours.Reject the null hypothesis. There is sufficient evidence to conclude that the lifetime of a bulb is less than 750 hours.

What conclusion would be appropriate for a significance level of 0.01?

Reject the null hypothesis. There is not sufficient evidence to conclude that the lifetime of a bulb is less than 750 hours.Reject the null hypothesis. There is sufficient evidence to conclude that the lifetime of a bulb is less than 750 hours.    Do not reject the null hypothesis. There is not sufficient evidence to conclude that the lifetime of a bulb is less than 750 hours.Do not reject the null hypothesis. There is sufficient evidence to conclude that the lifetime of a bulb is less than 750 hours.

What significance level and conclusion would you recommend and why?

Answer 1

To test against

Now,

The value of the test statistic = z_obs = -1.92

and P-value = 0.028

At 5% level of significance,

Since P-value < 0.05, so we reject H₀ at 5% level of significance.

Conclusion : Reject the null hypothesis. There is sufficient evidence to conclude that the lifetime of a bulb is less than 750 hours.

At 1% level of significance,

Since P-value > 0.01, so we fail to reject H₀ at 1% level of significance.

Conclusion : Do not reject the null hypothesis. There is not sufficient evidence to conclude that the lifetime of a bulb is less than 750 hours.

One should recommend 5% significance level since size = 5% is good enough as it controls type II errors well while having a decent value of type I error/size.