Chapter 10 review
This chapter primarily serves to introduce terminology. The actual calculations will be done slightly differently when we get to subsequent chapters. Thus, for this chapter what is important is to understand the meaning of terms such as confidence level, confidence interval, significance level, and so forth.
| Concept | Symbol | Definition | Example |
|---|---|---|---|
| Unknown Parameter | m | A characteristic of a population that we would like to estimate | The mean number of children per household for all families that live in zip code 20902 |
| Sample Statistic | A statistic used to estimate a parameter | The average number of children per household from a sample of a 100 families in the zip code. Suppose that this average is 3.3 with a standard deviation of 1.8 | |
| Confidence Interval | + or - m | A range of values around the sample statistic that is likely to contain the true value of the unknown parameter. | 3.3 + or - .36 |
| Margin of Error | m | The distance from the sample statistic to the end of the confidence interval | .36 |
| Confidence Level | C | The percentage of the time that the confidence interval would contain the true value of the parameter if this procedure were followed many times. We choose a confidence level. The margin of error then is derived based on the chosen confidence level, the standard deviation, and the sample size. | 95 percent |
| Critical Value of z | z* | The number of standard deviations away from that is the boundary of the confidence interval. To find z*, convert C to a decimal--call this c--and take invnorm(c/2) and invnorm (1-c/2). | -1.96, 1.96 |
| Sample Size | n | The sample size affects the width of the confidence interval. This is because z* = (+m)/(s/[square root of n]) | 100 |
| Null Hypothesis | H0 | A hypothesis about the value of an unknown parameter. We set up our test so that strong evidence is required to reject the null hypothesis. | Our null hypothesis is that the number of children per household in the zip code is equal to a national average of 1.8 |
| Alternative Hypothesis | Ha | A hypothesis that we would accept only if there were strong evidence in its favor. | Families in this zip code have more than 1.8 children each on average. |
| Two-sided alternative | Ha | An alternative hypothesis that the unknown parameter is either less than or greater than the value of the null hypothesis. | Families in this zip code either have more or less than 1.8 children each on average. |
| P-value | P | Probability that we would observe if the null hypothesis were true. To calculate this, let m = H0 and calculate z = ( - m)/(s/[square root of n]) and use normcdf() to evaluate the probability of z. | (something very small in this case, like .00001) |
| Type I error | I | Rejecting the null hypothesis when in fact it is true | The true mean number of children per family in our zip code is 1.8, but we reject the hypothesis that it is 1.8 |
| Type II error | II | Failing to reject the null hypothesis when in fact it is false | The true mean number of children per family in our zip code is greater than 1.8, but we fail to reject the hypothesis that it is 1.8 |
| Significance | a | The probability that our method would result in type I error. We choose a level of significance. | .05 |
We reject the null hypothesis when the P-value is less than the significance level. That is, we reject the null hypothesis when there is a sufficiently low probability that our test statistic would have been observed if the null hypothesis were true.
When we have a two-sided alternative hypothesis, there is a direct relationship between a confidence interval and the significance level. When the null hypothesis for m is outside of the confidence interval for a confidence level of, say, 90 percent, then we can reject the null hypothesis at a significance level of (1-c) or .10When we have a one-sided alternative hypothesis and a significance level of .10, then we do not need to go out as far into the tail to reject the null hypothesis. So, we use a narrower confidence interval, in this case 80 percent, to set the boundary for acceptance or rejection.