Complete Question
The quality control manager at a light bulb factory needs to estimate the mean life of a large shipment of light bulbs. The standard deviation is 100 hours. A random sample of 64 light bulbs indicated a sample mean life of 350 hours
A. Construct a 95% confidence interval estimate for the population mean life of light bulbs in this shipment.
B. Do you think that the manufacturer has the right to state that the light bulbs have a mean life of 400
a. No.
b. Yes.
c. Maybe.
d. Do not know
Answer:
A
[tex]325.5 < \mu < 374.5[/tex]
B
correct option is a
Step-by-step explanation:
From the question we are told that
The standard deviation is [tex]\sigma = 100[/tex]
The sample mean is [tex]\= x = 350 \ hours[/tex]
The sample size is [tex]n = 64[/tex]
Given that the confidence level is 95% then the level of significance is mathematically represented as
[tex]\alpha = (100- 95)\%[/tex]
[tex]\alpha = 0.05[/tex]
The critical value of [tex]\frac{\alpha }{2}[/tex] from the normal distribution table is
[tex]Z_{\frac{\alpha }{2 } } = 1.96[/tex]
Generally the margin of error is mathematically represented as
[tex]E = Z_{\frac{\alpha }{2} } * \frac{\sigma }{\sqrt{n} }[/tex]
=> [tex]E = 1.96 * \frac{100 }{\sqrt{64} }[/tex]
=> [tex]E = 24 .5[/tex]
Generally the 95% confidence interval is mathematically represented as
[tex]\= x - E < \mu < \= x + E[/tex]
[tex]350 - 24.5 < \mu < 350 + 24.5[/tex]
[tex]325.5 < \mu < 374.5[/tex]
This 95% confidence interval can be interpreted as follows
There is 95 confidence that the true population mean lies in this interval
Now looking at the interval we see that it dose not contain 400 hence
the correct answer is No the manufacturer do not have the right to state that the light bulbs last on average 400 hours
