Construct an $80\%$ confidence interval for the population mean given the sample mean $\stackrel{}{x}=8\mathrm{.}5,$ the population standard deviation $=0\mathrm{.}7,$ and a sample size $n=46.$

In a random sample of $28$ people, the mean commute time to work was $33\mathrm{.}8$ minutes with a sample standard deviation of $7\mathrm{.}2$ minutes. Assume the population was normally distributed and use a t$-$distribution to construct a $95\%$ confidence interval for the population mean?

An environmentalist estimates that the mean waste recycled by adults is more than $1$ pound per day. You want to test this claim. You find that the mean waste recycled per person per day for a random sample of $8$ adults in the country is $1\mathrm{.}1$ pounds and the standard deviation of the population is $0\mathrm{.}2$ pounds. At $=0\mathrm{.}05,$ can you support this claim? Assume the population is normally distributed.

a$)$ State the claim as a mathematical statement

b$)$ State the null hypothesis $H_0$

c$)$ State the alternate hypothesis $H_a$

d$)$ State the test statistic

e$)$ Do you reject $H_o$ or fail to reject $H_0?$

f$)$ Can you support the claim?

The earnings per share $($in dollars$)$ and the dividends per share $($in dollars$)$ for $6$ medical supply companies in a recent year are shown in the data set below.

$\backslash$table$[[$Earnings per share, $x,2\mathrm{.}78,5\mathrm{.}08,4\mathrm{.}51,3\mathrm{.}05,3\mathrm{.}78,2\mathrm{.}25],[$Dividends per share, $y,0\mathrm{.}57,2\mathrm{.}42,1\mathrm{.}46,0\mathrm{.}82,1\mathrm{.}07,0\mathrm{.}28]]$

a$)$ Determine the correlation coefficient $r=$

b$)$ Does this represent a positive correlation or a negative correlation?

c$)$ State a model linear equation, hat$(y)=mx+b,$ for the regression line.

d$)$ Use the model to predict the dividends per share for an earning share of $4.$