(10 points) a market research firm supplies manufacturers with estimates of the retail sales of their products from samples of retail stores. marketing managers are prone to look at the estimate and ignore sampling error. a random sample of 36 stores this year shows mean sales of 75 units of a small appliance with a standard deviation of 6 units. during the same point in time last year, a random sample of 49 stores had mean sales of 68 units with standard deviation 7 units. it is of interest to construct a 95 percent confidence interval for the difference in population means −, where is the mean of this year's sales and is the mean of last year's sales. enter values below rounded to three decimal places. (a) the estimate is: . (b) the standard error is: .

a)

b)  

On this case 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

  represent the sample mean 2

n1=36 represent the sample 1 size  

n2=49 represent the sample 2 size  

sample standard deviation for sample 1

sample standard deviation for sample 2

parameter of interest.

Part a

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

  (1)

The point of estimate for is just given by:

In order to calculate the critical value we need to find first the degrees of freedom, given by:

Since the 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: "=-T.INV(0.025,83)".And we see that

Part b

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    

step-by-step explanation: