Answer to Question #177002 in Statistics and Probability for Michael Faustino

Question

A factory manufacturing light emitting diode LED bulbs claims that their light bulbs last

for 50,000 hours on the average. To confirm if this is valid, a quality control manager obtained a

sample of 50 LED light bulbs and got a mean lifespan of 40,000 hours. The standard deviation of

the manufacturing process is 1000 hours.

1. What is the best point estimate for the true mean life span of the LED light bulbs

manufactured by this factory.

2. What is the standard error of this point estimate?

3. What is the margin of error of this point estimate?

4. Construct a 95% confidence interval of the true mean life span of LED light bulbs

manufactured by this factory.

5. Do you think that the claim of the manufacturer is valid? Explain.

6. If you want to be 99% confident that the point estimate is at most 100 hours from the true

mean life span, how many light bulbs should be included in the sample?

Answer 1

1.The best point estimate for the population mean is the sample mean.

2.

$SE=\sigma/\sqrt{n}=1000/\sqrt{50}=141.42$

3.

$ME=z_{\alpha}\sigma/\sqrt{n}$

$z$ - z-score consistent with confidence interval

4.

$Z_{0.025}<\frac{\overline{x}-\mu}{\sigma/\sqrt{n}}<Z_{0.975}$

$-1.96<\frac{\overline{x}-\mu}{\sigma/\sqrt{n}}<1.96$

$\overline{x}-1.96\sigma/\sqrt{n}<\mu<\overline{x}+1.96\sigma/\sqrt{n}$

$39722.81<\mu<40277.19$

5.The claim of the manufacturer is not valid. Claimed $\mu=50000$ is not in a 95% confidence interval.

6.

$Z_{0.995}=3.28=\frac{100}{\sigma/\sqrt{n}}$

$n=(3.28\cdot1000/100)^2=1075$