Q$9.$ The tensile strength of a certain type of thread is approximately normally distributed with standard deviation of $6\mathrm{.}8$ kilograms. A sample of $20$ pieces of the thread has an average tensile strength of $72\mathrm{.}8$ kilograms. Then,

$($a$)$ A point estimate of the population mean of the tensile strength $(m)$ is:

$($A$)72\mathrm{.}8$

$($B$)20$

$($C$)6\mathrm{.}8$

$($D$)46\mathrm{.}24$

$($E$)$

None of these

$($b$)$ Suppose that we want to estimate the population mean $(m)$ by the sample mean ${?}^{(\stackrel{}{x})}.$ To be $95\%$ confident that the error of our estimate of the mean of tensile strength will be less than $3\mathrm{.}4$ kilograms, the minimum sample size should be at least:

$($A$)4$

$($B$)16$

$($C$)20$

$($D$)$

$18$

$($E$)$ None of these

$($c$)$ For a $98\%$ confidence interval for the mean of tensile strength, we have the lower bound equal to:

$($A$)68\mathrm{.}45$

$($B$)69\mathrm{.}26$

$($C$)71\mathrm{.}44$

$($D$)69\mathrm{.}68$

$($E$)$ None of these

$($d$)$ For a $98\%$ confidence interval for the mean of tensile strength, we have the upper bound equal to:

$($A$)74\mathrm{.}16$

$($B$)77\mathrm{.}15$

$($C$)75\mathrm{.}92$

$($D$)76\mathrm{.}34$

$($E$)$ None of these

Solution:

a$)$

b$)$ c&d$)$

An electrical firm manufactures light bulbs that have a length of life that is approximately $10$ normally distributed with a standard deviation of $40$ hours. If a sample of $30$ bulbs has an average life of $780$ hours, find a $96\%$ confidence interval for the population mean of all bulbs produced by this firm.