Answer:
95% confidence interval for the mean breaking strength of the new steel cable is [763.65 lb , 772.75 lb].
Step-by-step explanation:
We are given that the engineers take a random sample of 45 cables and apply weights to each of them until they break. The mean breaking weight for the 45 cables is 768.2 lb. The standard deviation of the breaking weight for the sample is 15.1 lb.
Since, in the question it is not specified that how much confidence interval has be constructed; so we assume to be constructing of 95% confidence interval.
Firstly, the Pivotal quantity for 95% confidence interval for the population mean is given by;
P.Q. = ~
where, = sample mean breaking weight = 768.2 lb
s = sample standard deviation = 15.1 lb
n = sample of cables = 45
= population mean breaking strength
Here for constructing 95% confidence interval we have used One-sample t test statistics as we don't know about population standard deviation.
<u>So, 95% confidence interval for the population mean, </u><u> is ;</u>
P(-2.02 < < 2.02) = 0.95 {As the critical value of t at 44 degree
of freedom are -2.02 & 2.02 with P = 2.5%}
P(-2.02 < < 2.02) = 0.95
P( <
<
) = 0.95
P( <
<
) = 0.95
<u>95% confidence interval for</u> = [
,
]
= [ ,
]
= [763.65 lb , 772.75 lb]
Therefore, 95% confidence interval for the mean breaking strength of the new steel cable is [763.65 lb , 772.75 lb].