Question

9. (10 pts.) A manufacturer of light bulbs claims that the average lifespan of their light bulbs is at least 1600 hours. A sample of 100 light bulbs finds an average lifetime of 1570 hours with a standard deviation of 120 hours. If a one-sided confidence interval is considered, is the claim justified if a 95% confidence level is used?

Answer 1

Step 1:

Ho: $\mu$ ≥ 1600 (Claim)

Ha: $\mu$ < 1600

Step 2: Test statistics

n = 100

sample mean = 1570

sample sd = 120

Assuming that the data is normally distributed and as sample size is 100, we will use z stat

$z = \frac{\bar{x} - \mu}{\sigma/\sqrt{n}} = \frac{1570 - 1600}{120/\sqrt{100}}$

z = -2.50

Step 3: Rejection region  

df = 99

alpha = 0.05

The z-critical value for a left-tailed test, for a significance level of α = 0.05 is

zc = −1.64

Step 4: Decision

As the z stat (-2.50) falls in the rejection region, we reject the Null hypothesis

Step 5: Conclusion

Hence we have sufficient evidence to believe that the average mean lifetime is less than 1600