Question 16

  • Notice the condition

    Test For Population mean (V) Population mean (V) Population proportion (p) Difference of two means (VI — "2) Difference of two means (VI — "2) Mean difference (paired data) Difference of two proportions Null Hypothesis P P Test Statistic (f—go) po (I—po ) n 2 01 02 2 2 (DI -D2 )-O p(l—p) Distribution z z z tdistribution with the smaller of n—l and z Use When Normal distribution or n \> 30; o known n < 30, and/or o unknown nD, 2 10 Both normal distributions, or 171, 30; q, known 171, < 30; and/or q, g unknown n < 30 pairs of data and/or o d unknown nD, 10 for each group

Question 18

  • P-value for two-sided t-test = P-value for one-sided t-test * 2

    P -value in one-sided and two-sided tests Ha: It \> is z) One-sided (one-tailed) test Two-sided (two-tailed) test Izi TO calculate the P-value for a two-sided test, use the symmetry of the normal curve. Find the P-value for a one-sided test and double it.

Question 21

A good sample is representative. This means that each sample point represents the attributes of a known number of population elements. Bias often occurs when the survey sample does not accurately represent the population. The bias that results from an unrepresentative sample is called selection bias. Some common examples of selection bias are described below. • Undercoverage. Undercoverage occurs when some members of the population are inadequately represented in the sample. A classic example of undercoverage is the Literary Digest voter survey, which predicted that Alfred Landon would beat Franklin Roosevelt in the 1936 presidential election. The survey sample suffered from undercoverage of low-income voters, who tended to be Democrats. How did this happen? The survey relied on a convenience sample, drawn from telephone directories and car registration lists. In 1936, people who owned cars and telephones tended to be more affluent. Undercoverage is often a problem with convenience samples. • Nonresponse bias. Sometimes, individuals chosen for the sample are unwilling or unable to participate in the survey. Nonresponse bias is the bias that results when respondents differ in meaningful ways from nonrespondents. The Literary Digest survey illustrates this problem. Respondents tended to be Landon supporters; and nonrespondents, Roosevelt supporters. Since only 25% of the sampled voters actually completed the mail-in survey, survey results overestimated voter support for Alfred Landon. The Literary Digest experience illustrates a common problem with mail surveys. Response rate is often low, making mail surveys vulnerable to nonresponse bias. • Voluntary response bias. Voluntary response bias occurs when sample members are self- selected volunteers, as in voluntary samples. An example would be call-in radio shows that solicit audience participation in surveys on controversial topics (abortion, affirmative action, gun control, etc.). The resulting sample tends to overrepresent individuals who have strong oplruons.

Response bias refers to the bias that results from problems in the measurement process. Some examples of response bias are given below. Leading questions. The wording of the question may be loaded in some way to unduly favor one response over another. For example, a satisfaction survey may ask the respondent to indicate where she is satisfied, dissatisfied, or very dissatified. By giving the respondent one response option to express satisfaction and two response options to express dissatisfaction, this survey question is biased toward getting a dissatisfied response. Social desirability. Most people like to present themselves in a favorable light, so they will be reluctant to admit to unsavory attitudes or illegal activities in a survey, particularly if survey results are not confidential. Instead, their responses may be biased toward what they believe is socially desirable.

Question 25

Population Samples

Question 31

C:\\6432CA65\\FE01530B-89BD-4F8B-A3E1-55F12080AD12\_files\\image131.png

= /Ρ2σ

Question 32

  • H0: There is no association between A and B / Variables are independent

  • H1: There is an association between A and B / Variables are dependent

  • P > α

    • We fail to reject the null hypothesis and do not have evidence to support the claim that there is an association between A and B

  • P < α

    • We reject the null hypothesis and do have evidence to support the claim that there is an association between A and B

Question 37

The probability of making a type I error is called the significance level and is denoted by the Greek letter, (alpha). The significance level should be chosen before data is collected. The probability of making a type Il error is denoted by the Greek letter, ß (beta). The quantity is called the power of the test and represents the probability of rejecting the null hypothesis when it is actually false, in other words, making the correct decision by rejecting the null hypothesis. The power of the test is the probability on not making a type Il error. Any value of may be chosen, although statisticians usually choose small values. Common values for include = 0.01 — willingness to make a type I error 1 % of the time = 0.05 — willingness to make a type I error 5% of the time = 0.10 — willingness to make a type I error 10% of the time The person performing the test chooses the significance level a. How much power you get from a test depends on the significance level. The value of is very difficult or impossible to calculate. The calculation of the value of ß is beyond the scope of the AP Statistics Test, but you need to know what it means. The power of a test (the probability of rejecting the null hypothesis when it is false) increases as the significance level also increases. A test performed at = 0.10 has more power than a test performed at a = 0.05. Increasing the signifi- cance level, a, to increase the power can be counterproductive, since it increases the risk of making a type I error. Decreasing also decreases the probability of rejecting the null hypothesis. This reduces the power of the test, which increases the probability of making a type Il error, ß. Choosing a properly requires finding a balance appropriate for the situation. Decision Fail to reject Ho Reject Ho Ho Is Actually True Correct decision Type I error, a Ho Is Actually False Type Il error, Correct decision, 1 —

Fai' 'о гед Но .70 ReiecT Но .65 Роне г .63

The following points should be emphasized: Power gives the probability of avoiding a Type Il error. • Power has a different value for different possible correct values of the population parameter; thus it is actually a function where the independent variable ranges over specific alternative hypotheses. Choosing a smaller a (that is, a tougher standard to reject Ho) results in a higher risk of Type Il error and a lower power—observe in the above graphs how mak- ing a smaller (in this case moving the critical cutoff value to the left) makes the power less and ß more\! The greater the difference between the null hypothesis A) and the true value p, the smaller the risk of a Type Il error and the greater the power—observe in the above picture how moving the lower graph to the left makes the power greater and ß less. (The difference between PO and p is sometimes called the effect—thus the greater the effect, the greater is the power to pick it up.) A larger sample size n will reduce the standard deviations, making both graphs narrower resulting in smaller smaller and larger power\!

results matching ""

    No results matching ""