$(35\%)$ During a dam construction project, the site engineer takes $9$ concrete samples a day for a

period of three months. The mean compressive strengths of these samples taken at the beginning of

the project $($Day$-1)$ in dry conditions are as given as follows $($in MPa$)$:

$40\mathrm{.}2,30\mathrm{.}4,28\mathrm{.}9,30\mathrm{.}5,22\mathrm{.}4,25\mathrm{.}8,18\mathrm{.}4,14\mathrm{.}2,15\mathrm{.}3.$

$(5\%)$ a$)$ Find the sample mean, variance, coefficients of variation, coefficients of skewness and

kurtosis for the above given sample.

$(5\%)$ b$)$ Find a $99\%$ lower one$-$sided confidence interval on true mean compressive strength.

$(5\%)$ c$)$ Find a $98\%$ two$-$sided confidence interval on true mean compressive strength.

$(5\%)$ d$)$ Find a $99\%$ upper one$-$sided confidence interval on the true variance of compressive strength.

$(5\%)$ e$)$ Suppose that the largest observation $40\mathrm{.}2$ was misrecorded and should actually be $20\mathrm{.}4.$

Compute the sample mean, $\stackrel{}{x},$ and the sample variance, $s^2.$ Repeat options $($c$)$ and $($d$)$ for these new

values. Compare the original computed intervals and the newly computed intervals with the corrected

observation value. Please quantify the effect of such a mistake on the values of the sample mean,

sample variance, and the width of the two$-$sided confidence interval.

$(5\%)$ f$)$ Taking option $($e$)$ into account, at the end of first month $($Days $1-30),$ it is seen that the sample

mean and variance of all the samples collected are $\stackrel{}{x}=22\mathrm{.}1$ and the sample variance $s^2=40\mathrm{.}5,$

respectively. Repeat the options $($c$)$ and $($d$)$ and compare the results.

$(5\%)g)$ The site engineer has recorded also the temperatures $($at $12$:$00$ PM everyday$)$ at the site in the

first week of construction $($given in table below$).$ Determine the dependent $(Y)$ and independent $(x)$

variables of the problem. Find the covariance and the correlation coefficient for the given data.

Determine the equation of the regression line $)=(widehat()+$widehat$()x$ and find $80\%$ confidence interval for the

slope and intercept of the regression line.

Table $1.$ Mean compressive strengths of concrete samples and temperatures for the first seven days.