A personal fitness produces both a deluxe and a standard...

A personal fitness produces both a deluxe and a standard model of a smoothie blender for home use. Selling prices obtained from a sample of retail outlets follow. Excel File: data10-27.xlsx Model Price ($) Model Price ($) Retail Outlet Deluxe Standard Retail Outlet Deluxe Standard 1 39 27 5 40 30 2 39 28 6 39 34 3 45 35 7 35 29 4 38 30 Round your answers to 2 decimal places. a. The manufacturer's suggested retail prices for the two models show a $10 price differential. Use a 0.05 level of significance and test that the mean difference between the prices of the two models is $10. Do not reject the null hypothesis (Ho=10) b. What is the 95% confidence interval for the difference between the mean prices of the two models? 5.63 to 6.31

Transcript

00:01 So for the first part of the problem here, the first thing that we want to do is find the differences between each one of the sample deluxe and standard values.

00:11 When we go through and calculate that out, we should get 12 for retail outlet 1, 11 for 2, 10 for 3, 8 for 4, 10 for 5, 5 for 6, and 6 for 7.

00:27 Then we want to find the mean value, or pardon me, the sample mean value of the differences, which we find by taking the sum of the differences and dividing it by the number of them, so we divide by 7.

00:39 When we do that, we'll find that the mean difference is equal to approximately 8 .86.

00:45 We also need to find the standard deviation of our differences, which we do by taking the sum of, or not some, pardon me, square root of one over our sample size minus 1, so 1 over 6, times the square root of the difference between each individual difference and the sample mean difference, 8 .86, squared.

01:08 When we do that, we'll find that our standard deviation is going to give us a value of 2 .61.

01:18 Actually, i'll just give a little bit more detail on the standard deviation calculation there.

01:22 So that's going to be, for instance, 1 over 6 times, i'll just show the first few terms, we do 12 minus 8 .86 squared, plus 11 minus 8 .86 squared, and so on.

01:42 So for those first couple of terms, 12 minus 8 .86, well, that's going to be 3 .14.

01:48 So this is square root of 1 over 6 times 3 .14 squared, plus 11 .14 squared plus 11 .1 .4 squared, plus 11 .5.

01:57 11 minus 8 .86 would be 2 .14, then we square that, and so on.

02:04 And we get the sample standard deviation of differences of 2 .61 at the end.

02:10 Now, at the alpha equals 0 .05, 0 .05 level of significance.

02:19 I'll note that this is going to be a two -tailed hypothesis test with the null hypothesis, that the population mean difference is equal to 10.

02:29 The alternate hypothesis that the population mean difference simply does not equal 10.

02:36 So we'll have that our critical z value, it's going to be z for a proportion of alpha over 2 in the tail, so 0 .025, which gives us a value of 1 .96, roughly...

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