Aron Chapter 6 (96%)
What is 95% confidence interval?
Confidence interval in which, roughly speaking, there is a 95% chance that the population mean falls within this interval. A 95% confidence interval is the range of values that you are 95% confident includes the population mean, estimated based on the scores in a sample. Aron (2010), C6, S210
What is 99% confidence interval?
Confidence interval in which, roughly speaking, there is a 99% chance that the population mean falls within this interval. Aron (2010), C6, S210
What is shape of a distribution of means?
Contour of a histogram of a distribution of means, such as whether it follows a normal curve or is skewed; in general, a distribution of means will tend to be unimodal and symmetrical and is often normal. Aron (2010), C6, S198
Rules and Formulas for Determining the Characteristics of a Distribution of Means?
Rule 1: The mean of a distribution of means is the same as the mean of the population of individuals: Rule 2a: The variance of a distribution of means is the variance of the population of individuals divided by the number of individuals in each sample: Rule 2b: The standard deviation of a distribution of means is the square root of the variance of the distribution of means: Rule 3: The shape of a distribution of means is approximately normal if either (a) each sample is of 30 or more individuals or (b) the distribution of the population of individuals is normal. Aron (2010), C6, S196
What is The standard deviation of a distribution of means (Population SDM )?
Square root of the variance of the distribution of means; same as standard error (SE). Aron (2010), C6, S198
A researcher predicts that showing a certain film will change people's attitudes toward alcohol. The researcher then randomly selects 36 people, shows them the film, and gives them an attitude questionnaire. The mean score on the attitude test for these 36 people is 70. The score on this test for people in the general population (who do not see the film) is 75, with a standard deviation of 12. (a) Find the best estimate of the mean of people in general who see the film and (b) its 95% confidence interval. (c) Compare this result to the conclusion you drew when you used this example in the "How are you doing?" section for hypothesis testing with a distribution of means.
(a) The best estimate is the sample mean: 70. (b) Population SD2M = Population SD2/N = 122/36 = 4. Thus, the standard deviation of the distribution of means, Population SDM = √Population SD2M = 24 = 2. The lower confidence limit = (-1.96)(2) + 70 = -3.92 + 70 = 66.08. the upper confidence limit = (1.96)(2) + 70 = 73.92. The 95% confidence interval is from 66.08 to 73.92. (c) The 95% confidence interval does not include the mean of the general population (which was 75). Thus, you can reject the null hypothesis that the two populations are the same. This is the same conclusion as when using this example for hypothesis testing. Aron (2010), C6, S212
(a) What is the best estimate of a population mean? (b) Why?
(a) The best estimate of a population mean is the sample mean. (b) It is more likely to have come from a population with the same mean than from any other population. Aron (2010), C6, S212
(a) What number is used to indicate the accuracy of an estimate of the population mean? (b) Why?
(a) The standard deviation of the distribution of means (or standard error) is used to indicate the accuracy of an estimate of the population mean. (b) The standard deviation of the distribution of means (standard error) is roughly the average amount that means vary from the mean of the distribution of means. Aron (2010), C6, S212
(a) What is the standard error (SE)? (b) Why does it have this name?
(a) The standard error is the standard deviation of the distribution of means. (b) It has this name because it tells you about how much means of samples typically (standardly) differ from the population mean, and thus tells you the typical amount that the means of samples are in error as estimates of the population mean. Aron (2010), C6, S202
What is confidence interval (CI)?
Roughly speaking, the region of scores (that is, the scores between an upper and lower value) that is likely to include the true population mean; more precisely, the range of possible population means from which it is not highly unlikely that you could have obtained your sample mean. Aron (2010), C6, S209
Explain how you could create a distribution of means by taking a large number of samples of four individuals each.
Take a random sample of four scores from the population and figure its mean. Do this a very large number of times. Make a distribution of all of the means. Aron (2010), C6, S202
What is confidence limit?
Upper or lower value of a confidence interval. Aron (2010), C6, S209
What is variance of a distribution of means?
Variance of the population divided by the number of scores in each sample. Aron (2010), C6, S197
How do you find the Z score for the sample's mean on the distribution of means?
You use the usual formula for changing a raw score to a Z score, using the mean and standard deviation of the distribution of means. The formula is Z = (M - Population MM)/Population SDM. Aron (2010), C6, S206
Rule 1 stated as a formula?
Population MM = Population M Aron (2010), C6, S197
Rule 2a stated as a formula?
Population SD2M = Population SD2 / N Aron (2010), C6, S199
Rule 2b stated as a formula?
Population SDM = √Population SD2M Aron (2010), C6, S199
What is distribution of means?
Distribution of means of samples of a given size from a particular population (also called a sampling distribution of the mean); comparison distribution when testing hypotheses involving a single sample of more than one individual. Aron (2010), C6, S194
What is Z test?
Hypothesis-testing procedure in which there is a single sample and the population variance is known. Aron (2010), C6, S202
How is hypothesis testing with a sample of more than one person different from hypothesis testing with a sample of a single person?
In hypothesis testing with a sample of more than one person, the comparison distribution is a distribution of means. Aron (2010), C6, S206
A population of individuals has a normal distribution, a mean of 60, and a standard deviation of 10. What are the characteristics of a distribution of means from this population for samples of four each?
The characteristics of a distribution of means are calculated as follows: Population MM = Population M = 60. Population SD2M = Population SD2 / N = 102/4 = 25; Population SDM = 5. Shape = normal. Aron (2010), C6, S202
Write the formula for the variance of the distribution of means, and define each of the symbols.
The formula for the variance of the distribution of means is: Population Population SD2M = Population SD2 / N .Population SD2M is the variance of the distribution of means; Population SD2 is the variance of the population of individuals; N is the number of individuals in your sample. Aron (2010), C6, S202
What is mean of a distribution of means?
The mean of a distribution of means of samples of a given size from a particular population; it comes out to be the same as the mean of the population of individuals. Aron (2010), C6, S196
a) Why is the mean of the distribution of means the same as the mean of the population of individuals? (b) Why is the variance of a distribution of means smaller than the variance of the distribution of the population of individuals?
a) With randomly taken samples, some will have higher means and some lower means than the population of individuals; in the long run these have to balance out. (b) You are less likely to get a sample of several scores that has an extreme mean than you are to get a single extreme score. This is because in any random sample it is highly unlikely to get several extremes in the same direction; extreme scores tend to be balanced out by middle scores or extremes in the opposite direction. Thus, with fewer extreme scores and more middle scores, there is less variance. Aron (2010), C6, S202
Steps for Figuring the 95% and 99% Confidence Intervals?
❶ Estimate the population mean and figure the standard deviation of the distribution of means. ❷ Find the Z scores that go with the confidence interval you want. ❸ To find the confidence interval, change these Z scores to raw scores. Aron (2010), C6, S211
A researcher predicts that showing a certain film will change people's attitudes toward alcohol. The researcher then randomly selects 36 people, shows them the film, and gives them an attitude questionnaire. The mean score on the attitude test for these 36 people is 70. The score for people in general on this test is 75, with a standard deviation of 12. Using the five steps of hypothesis testing and the 5% significance level, carry out a Z test to see if showing the film changed people's attitudes toward alcohol.
❶ Restate the question as a research hypothesis and a null hypothesis about the populations. The two populations are these: Population 1: People shown the film. Population 2: People in general (who are not shown the film). The research hypothesis is that the mean attitude of the population shown the film is different from the mean attitude of the population of people in general; the null hypothesis is that the populations have the same mean attitude score. ❷ Determine the characteristics of the comparison distribution. Population MM = Population M = 75. Population SD2M = Population SD2 / N = 122/36 = 4. Population SDM = 2. Shape is normal. ❸ Determine the cutoff sample score on the comparison distribution at which the null hypothesis should be rejected. Two-tailed cutoffs, 5% significance level, are +1.96 and -1.96. ❹ Determine your sample's score on the comparison distribution. Z = (M - Population MM)/Population SDM = (70 - 75)/2 = -2.50. ❺ Decide whether to reject the null hypothesis. The Z score of the sample's mean is -2.50, which is more extreme than -1.96; reject the null hypothesis. Seeing the film does change attitudes toward alcohol. Aron (2010), C6, S206
