Hypothesis Tests: One Sample Mean Chapter 12 Hypothesis Tests: One Sample Mean

Major Points An example Sampling distribution of the mean Testing hypotheses: Factors affecting the test Cont.

An Example :Media Violence Does violent content in a video affect subsequent responding? Same example: 100 subjects saw a video containing considerable violence. Then free associated to 26 homonyms that had an aggressive & nonaggressive form. e.g. cuff, mug, plaster, pound, sock Cont.

Media Violence--cont. Results Mean number of aggressive free associates = 7.10 Assume we know that without aggressive video the mean would be 5.65, and the standard deviation = 4.5 in population. These are parameters (m and s) Is 7.10 enough larger than 5.65 to conclude that video affected results?

Sampling Distribution of the Mean We need to know what kinds of sample means to expect if video has no effect. i. e. What kinds of means if m = 5.65 and s = 4.5? This is the sampling distribution of the mean. Cont.

Sampling Distribution of the Mean--cont. In Chapter 8 we saw exactly what this distribution would look like. It is called Sampling Distribution of the Mean. Why? See next slide. Cont.

Note that the SD of the sampling distribution is smaller than the SD of the population. It’s called the standard error, and we will see the formula for this later on. Cont.

Sampling Distribution of the Mean The sampling distribution of the mean depends on Mean of sampled population St. dev. of sampled population: Size of samples Larger sample sizes drawn, sampling distribution tends to be more normally distributed. Cont.

Sampling Distribution of mean Shape of the sampled population Approaches normal when population from which samples are drawn is normal Rate of approach depends on sample size Basic theorem Central limit theorem: states that the sampling distribution of means from RANDOM samples approaches a normal distribution regardless of the shape of the parent population. Additional readings

Central Limit Theorem Given a population with mean = m and standard deviation = s , the sampling distribution of the mean (the distribution of sample means) has a given mean and a standard deviation (we can figure these out). The distribution approaches normal as n, the sample size, increases.

Demonstration Let population be very skewed Draw samples of 3 and calculate means Draw samples of 10 and calculate means Plot means Note changes in means, standard deviations, and shapes Cont.

Parent Population Cont.

Sampling Distribution n = 3 Cont.

Sampling Distribution n = 10 Cont.

Demonstration--cont. Means have stayed at 3.00 throughout--except for minor sampling error Standard deviations have decreased appropriately Shapes have become more normal--see superimposed normal distribution for reference

Testing Hypotheses: s known H0: m = 5.65 H1: m  5.65 (Two-tailed alternate H) Calculate p(sample mean) = 7.10 if m = 5.65 Use z from normal distribution Sampling distribution would be normal

Using z To Test H0 : we can use the properties of normal curve to test hypotheses Calculate z = of the means, or the standard error of the mean, population SD/ square root of n, sample size If z > + 1.96, reject H0- remember that 1.96 leaves 5% in each tail, 95% between 2 SD’s Z value of 3.22 > 1.96 The difference is significant. Cont.

z--cont. Compare computed z to histogram of sampling distribution The results should look consistent. Logic of test Calculate probability of getting this mean if null true. Reject if that probability is too small. Choose an alpha, usually .05, but also .01 or .001.

Testing When s Not Known Assume same example, but s not known Can’t substitute s for s because s more likely to be too small See next slide. Do it anyway, but call answer t Compare t to tabled values.

Degrees of Freedom Skewness of sampling distribution of variance decreases as n increases t will differ from z less as sample size increases Therefore need to adjust t accordingly for sample size df = n - 1 t based on df

t Distribution- see Table D.6 or pg 292 Notice if you use a one-tailed test at alpha = .05, you don’t need as large of a CV to reject the null

Conclusions With n = 100, t.0599 = 1.98 (from a t table) Because t = 3.22 > 1.98, reject H0 (to calculate t use formula pg 288, where t = difference in means/SEmean calculated from the sample sd, or 7.10-5.65 / 4.5/sq root of 100 (100 is sample size, 4.5 was sd from the sample) Note that t and z are nearly identical in this case, in other cases the sample sd may not be a completely accurate estimate of pop sd Conclude that viewing violent video leads to more aggressive free associates than normal.

Factors Affecting t Difference between sample and population means Magnitude of sample variance Sample size

Factors Affecting Decision Significance level a One-tailed versus two-tailed test