Pediatricians claim that 3rd graders follow a normal distribution with a mean of 50 inches. You believe that the average height is less. So you want to test this out.

Ha (Alternative Hypothesis): Average height of 3rd graders < 50

H0 (Null Hypothesis): Average height of 3rd graders = 50

To do so, you take a sample of 30, 3rd graders. This sample has a mean of 47 inches, and a standard deviation of 2.1 inches?

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Now that you have successfully identified the distribution for the population (of all samples of size 30 that you could draw from within the 3rd graders), place your sample on this distribution:

Based on where your sample is placed in this distribution, we will calculate a few more statistics:

Q10. Now standardize this distance by dividing it by the POPULATION standard-deviation (or standard error). In other words, how many population-standard-deviations (or standard-error) away is your sample mean from the population mean?

-5.1

-9.1

-7.8

-3.4

-0.3

Q11. In the previous question, why did you divide the absolute distance (i.e., x - u) by the population standard deviation (or Standard Error) and not the sample standard deviation?

a. Because you're calculating how far is your sample-mean from the population mean on the population distribution

b. Because you're calculation how far is your sample-mean from the population mean on the sample distribution.

Q12. What is the standardized distance that you calculated in the previous question, called.

Hint: It's a test-statistic

a. t-value

b. z-score

c. significance level

d. p-value

Q13. What does z-score mean?

a. It is the inches your sample's mean is less than your population mean

b. It is the number of standard deviations away the sample mean is from 0.

c. It is the number of population- standard deviations (or standard error) away the sample mean is from the assumed population-mean, assuming the null hypothesis is true.

d. It is the inches your sample's mean is more than your population mean

Q14. If your z-score was +3, you would say you lie 3 standard deviations to the right of the mean; similarly, if you z-score was -3, you would say you lie -3 standard deviation away.

Now, most of the data (99% or more) lies within this range -3 standard deviations to +3 standard deviations from the data (which is why you may have heard of the word 6-sigma being used in supply-chain/logistics context.)

So, what does it mean when you find a z-score which is more negative that -3 standard deviations?

a. That your sample is on the very extreme left of the distribution, and the likelihood of such a sample being picked is very very small.

b. That your sample is on the very extreme right of the distribution, and the likelihood of such a sample being picked is very very small.

Q15. Why do we standardize the absolute distance 'x - u' (i.e., why do we divide the distance 'x - u' by the population standard-deviation - or SE) to calculate the test-statistic?

a. To standardize the distance, so that it is NOT expressed in units, such as gallons, or inches (i.e., the units in which the DV is measured) but in terms of (population) standard-deviations (i.e., how many standard deviations away is the sample-mean from the population mean). This standardized difference is our test statistic. By calculating the test-statistic in such a way, we can make sure that scale of DV has no effect on the test statistic EXAMPLE- finding a sample of 3rd graders that are 3 inches shorter than 50 inches (i.e., the population mean under the null), or finding a sample of giants in Harry Potter that are 3 feet shorter than the mean height of all giants (i.e., population mean), is likely to be just as rare. However, if we took the unstandardized difference in heights (i.e., simply x-u), it would seem that the sample of giants are more distinct from the population mean, while the sample of 3rd graders aren't as distinct from the population mean. That is why we standardize!

b. Because statisticians like to make things complicated!

Q16. Without using the calculator to calculate p-value, what do you think the p-value might be, looking at the z-score (that you calculated in Q 10).

a. small, because the z score indicates that the sample mean is very distant from the population-mean-under-the-null

b. big, because the z score indicates that the sample mean is very distant from the population-mean-under-the-null

c. small, because the z score indicates that the sample mean is very close to the population-mean-under-the-null

d. big, because the z score indicates that the sample mean is very close to the population-mean-under-the-null