Each person in random samples of 227 male and 293 female working adults living in a certain town in canada was asked how long, in minutes, his or her typical daily commute was. (use a statistical computer package to calculate the p-value. use μmales − μfemales. round your test statistic to two decimal places, your df down to the nearest whole number, and your p-value to three decimal places.)

the 95% confidence interval would be given by  

Step-by-step explanation:

Previous concepts  

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".  

The margin of error is the range of values below and above the sample statistic in a confidence interval.  

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".  

represent the sample mean 1  (males)

represent the sample mean 2  (females)

n1=227 represent the sample 1 size (males)  

n2=293 represent the sample 2 size (females)

sample standard deviation for sample 1  (males)

sample standard deviation for sample 2 (females)

represent the population standard deviation

parameter of interest.

Confidence interval

The confidence interval for the difference of means is given by the following formula:  

(1)  

The point of estimate for is just given by:

Since the sample size is large enough we can assume that th t distirbution is approximately equal to the normal distribution in order to find the quantile.

Let's assume a Confidence is 0.95 or 95%, the value of and , and we can use excel, a calculator or a tabel to find the critical value. The excel command would be: "=-NORM.INV(0.025,0,1)".And we see that  

The standard error is given by the following formula:

And replacing we have:

Confidence interval

Now we have everything in order to replace into formula (1):  

So on this case the 95% confidence interval would be given by  

R code

> barmale=30.2

> barfemale=27.9

> diff=barmale-barfemale

> smale=24

> sfemale=24.3

> nmale=227

> nfemale=293

> SE=sqrt((smale^2)/nmale +(sfemale^2)/nfemale)

> ME=qnorm(1-0.025)*SE

> lower=diff-ME;lower

[1] -1.882018

> upper=diff+ME;upper

[1] 6.482018

And we got the same results.