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.
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