Stats - Chapter 8
What is the reason the significance level of a test is not made arbitrarily small?
Doing so decreases the probability of correctly rejecting the null hypothesis The significance level is the probability of rejecting the null hypothesis when, in fact, the null hypothesis is true. While a small significance level is desireable, decreasing the significance level comes with the trade-off of also decreasing the probability of rejecting the null hypothesis when it should be rejected, that is, when the null hypothesis is false.
Which of the following best describes hypotheses? Statements about sample statistics Statements that cannot be tested Statements that are always true Statements about population parameters
Statements about population parameters
In hypothesis testing, what does an extreme value for the test statistic indicate?
The null hypothesis is not true A test statistic of zero means that the value of the sample statistic is exactly the same as the hypothesized value of the population parameter, so an extreme value for the test statistic contradicts the null hypothesis.
In hypothesis testing, what does a negative test statistic mean?
The sample proportion was less than the assumed population proportion in the null hypothesis The test statistic is calculated by subtracting the assumed population proportion in the null hypothesis from the sample proportion, then dividing by a positive number. If the sample proportion is less than the assumed population proportion, the test statistic will be negative.
For each graph, indicate whether the shaded area could represent a p-value. Explain why or why not. If yes, state whether the area could represent the p-value for a one-tailed or a two-tailed alternative hypothesis. (A) A Normal curve is plotted on a horizontal axis marked from less than negative 3 to 3 plus in intervals of 1 and centered at 0. Two vertical lines are plotted at approximate horizontal coordinates negative 1.5 and 1.5. The area below the curve to the left of the vertical line at negative 1.5 and the area below the curve to the right of the vertical line at 1.5 are shaded.
The shaded area could be a p-value for a test with a two-tailed alternative hypothesis since both tails are of equal size. Assuming the null hypothesis is true, the p-value is the probability of a test statistic being as extreme or more extreme than the one that was found. In a normal distribution (like the one shown) the mean is in the middle and is equal to the least extreme result. Values that are more extreme than the value of the test statistic are further from the mean than the test statistic.
A Normal curve is plotted on a horizontal axis marked from less than negative 3 to 3 plus in intervals of 1 and centered at 0. Two vertical lines are plotted at approximate horizontal coordinates negative 1.5 and 1.5. The area below the curve between the vertical lines is shaded.
The shaded area could not be a p-value because it does not include tail areas. Assuming the null hypothesis is true, the p-value is the probability of a test statistic being as extreme or more extreme than the one that was found. In a normal distribution (like the one shown) the mean is in the middle and is equal to the least extreme result. Values that are more extreme than the value of the test statistic are further from the mean than the test statistic. Since the shaded region includes values that are less extreme than the test statistic, this graph's shaded region does not represent a p-value.
A psychologist is interested in testing whether offering students a financial incentive improves their video-game-playing skills. She collects data and performs a hypothesis test to test whether the probability of getting to the highest level of a video game is greater with a financial incentive than without. Her null hypothesis is that the probability of getting to this level is the same with or without a financial incentive. The alternative is that this probability is greater. She gets a p-value from her hypothesis test of 0.003. What is the best interpretation of the p-value?
The p-value is the probability of getting a result as extreme as or more extreme than the one obtained, assuming that financial incentives are not effective in this context The p-value is a probability. Assuming the null hypothesis is true, the p-value is the probability that if the experiment were repeated, you would get a test statistic as extreme as or more extreme than the one you actually got. A small p-value suggests that a surprising outcome has occurred and discredits the null hypothesis.
In hypothesis testing, when should the null hypothesis be rejected?
When the p-value is less than the significance level The p-value is the probability that if the null hypothesis is true, a test statistic will have a value as extreme as or more extreme than the value observed. The significance level is the probability of rejecting the null hypothesis when, in fact, the null hypothesis is true. If the p-value is less than the significance level, there is sufficient evident to support the alternative hypothesis and the null hypothesis is rejected.
