SOLUTION: A light bulb company claims that the 40 watt light bulb it sells has an average life of 1000 hours with a standard deviation of 75 hours. Sixty-four (64) new bulbs were allowed to

Question 1167946: A light bulb company claims that the 40 watt light bulb it sells has an average life of 1000 hours with a standard deviation of 75 hours. Sixty-four (64) new bulbs were allowed to burn out to test this claim. The average lifetime of these bulbs was found to be 975 hours.Does this indicate that the average life of a bulb is not 1000 hours?

(Use a 5% level of significance)

Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
mean of population is assumed to be 1000
standard deviation of population is known to be 75.

sample size is 64
mean = 975
standard error = standard deviation / sqrt(sample size) = 75/sqrt(64) = 75/8 = 9.375

two tailed 5% level of significance level means alpha = .025 on each end of the confidence interval.

z-score associated with that is z = plus or minus 1.96.

z-score of distribution of sample means is z = (975 - 1000) / 9.375 = -2.67.

2.67 is greater than 1.96.
this indicates that the sample mean is outside the confidence interval meaning that the claim of 1000 hours life for the bulbs is rejected as not being a true reflection of the average life of the bulbs.

if the z-score of the mean of the sample was between the confidence interval limits of plus or minus 1.96, then it would have been considered just random variations in the mean of samples with size of 64 and the claim of an average of 1000 hours of life for the bulbs would have been accepted.

here's a reference that talks about the difference between standard deviation and standard error.

https://www.greenbook.org/marketing-research/how-to-interpret-standard-deviation-and-standard-error-in-survey-research-03377

this is what the mean of the sample looks like at 95% confidence level.

plus or minus 1.96 were the critical z-scores of the test.
since -2.67 was outside of these limits, then the null hypothesis was rejected.

if you worked by alpha, then the alpha was .025.
this was the area under the normal distribution curve that was not acceptable.
the alpha of the test score of -2.67 was .00379.
since this was less than the critical alpha of .025, then the null hypothesis was rejected.

the null hypothesis was that the population mean was 1000.
the alternate hypothesis was that the population mean was not 1000.
the sample mean, being outside the confidence limits, indicated that the population mean was, in all probability, not 1000.
RELATED QUESTIONS
1. Liwanag Bulb Company claims that it sells light bulbs with an average life of 1000... (answered by Theo)

A 100-watt light bulb has an average brightness of 1650 lumens, with a standard deviation (answered by Boreal)

a manufacturer of light bulbs claims that its new light bulb has a mean life of 1000... (answered by stanbon)

A light bulb manufacturer claims his light bulbs will last 500 hours on the average. The... (answered by stanbon)

In a certain city, electricity costs $0.10 per kW�h. What is the annual cost for... (answered by ikleyn,MathTherapy)

A new type of light bulb has an average lifespan of 1200 hours with a standard deviation... (answered by ewatrrr)

Light Bulb Life Span � The life span of a certain type of light bulb is normally... (answered by stanbon)

a manufacturer of light bulbs claims that its new light bulb has a mean life of 1,000... (answered by Fombitz)

Light bulbs are measured in lumens (light output), watts (energy used), and hours (life). (answered by stanbon)