I have been collecting pennies for decades. I have a 55 gallon barrel full of those pennies. Because I use the barrel for exercise and during that exercise I roll and even flip the barrel I know that the pennies are thoroughly mixed up. I have a coin collector who wants to buy my collection, but only if the average of the years on the pennies is less than 1968. I am sure that such is the case. However, I want to be really sure. It would take too much time and effort to go through and analyze the entire collection. Therefore, I take a random sample of 32 pennies and I look at the years on those coins.

I calculate the

Having seen, as we did in earlier pages, that the

To do this we look at the possibilities and we formulate a hypothesis. We will look at three possibilities about the average year of coins in the population. The average can be less than 1968, it can be equal to 1968, or it can be greater than 1968. Admittedly, the middle choice, equal to 1968, is almost certainly not true just because it would be such a coincidence to have the sum of all the years be exactly 1968 times the number of coins in the population.

Still, that middle choice is much more helpful than is either other choice. The middle choice gives us a solid figure, namely, 1968. The other choices merely say that the true value is not 1968; they do not give us a specific value that we can use. We formulate two hypotheses, H

We have our sample. We have calculated the

Table 2 | ||

The Truth (reality) | ||

Our Action | H_{0} is TRUE |
H_{0} is FALSE |

Reject H_{0} |
This is a Type I error |
made the correct decision |

Do not Reject H _{0} |
made the correct decision |
This is a Type II error |

A

A

Thus we have two kinds of errors,

For now, just looking at our example, there is not enough information for us to come out with a good rule on when to

Consider two cases. In the first case we happen to know that the

On the other hand, if the standard deviation of the coin years in the population is 45 years, then the standard deviation of

Finally, let us look at the case where the standard deviation of the coin years in the population is 21. Then the standard deviation of

If we are not willing to make a

The reason that we cannot get a good measure of

©Roger M. Palay Saline, MI 48176 December, 2015