Chapter 12Hypothesis Tests: One

Sample Mean

2

Major Points

• An example• Sampling distribution of the mean• Testing hypotheses:

– An example• Factors affecting the test

Cont.

3

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.

4

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?

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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.

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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.

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Mean Number Aggressive Associates

7.257.00

6.756.50

6.256.00

5.755.50

5.255.00

4.754.50

4.254.00

3.75

Sampling Distribution

Number of Aggressive AssociatesF

req

ue

nc

y

1400

1200

1000

800

600

400

200

0

Std. Dev = .45

Mean = 5.65

N = 10000.00

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.

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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.

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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

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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.

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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.

12

X

20.018.0

16.014.0

12.010.0

8.06.0

4.02.0

0.0

Skewed Population F

requ

ency

3000

2000

1000

0

Std. Dev = 2.43

Mean = 3.0

N = 10000.00

Parent Population

Cont.

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Sampling Distribution n = 3

Sample Mean

13.0012.00

11.0010.00

9.008.00

7.006.00

5.004.00

3.002.00

1.000.00

Sampling Distribution

Sample size = n = 3F

requ

ency

2000

1000

0

Std. Dev = 1.40

Mean = 2.99

N = 10000.00

Cont.

14

Sampling Distribution n = 10

Sample Mean

6.506.00

5.505.00

4.504.00

3.503.00

2.502.00

1.501.00

Sampling Distribution

Sample size = n = 10F

requ

ency

1600

1400

1200

1000

800

600

400

200

0

Std. Dev = .77

Mean = 2.99

N = 10000.00

Cont.

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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

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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

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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.

22.345.

45.1

45.

65.51.7

X

X

z

Cont.

X

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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.

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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.

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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

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t Distribution- see Table D.6 or pg 292One-Tailed Test

.05 .025 Two-Tailed Test

df .10 .05

10 1.812 2.228 15 1.753 2.131 20 1.725 2.086 25 1.708 2.060 30 1.697 2.042 100 1.660 1.984

Notice if you use a one-tailed test at alpha = .05, you don’t need as large of a CV to reject the null

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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.

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Factors Affecting t

• Difference between sample and population means

• Magnitude of sample variance• Sample size

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Factors Affecting Decision

• Significance level a• One-tailed versus two-tailed test