• The process of using the table to translate z-scores into portions of the total area can be reversed to calculate score values from particular portions of area or percentages.

• What if we wanted to know the salary level that defines the top 10% of earners?

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2

• The threshold for the top 10% can be determined by entering the table for a particular portion of area (such as the highest 10% tail) to determine the z-score values and then translating that z score into its raw score (salary) equivalent.

• Specifically, we scan through the table to locate the area beyond z that is closest to 10%.

3

• The table shows indicates that 10.03% falls beyond a z-score of 1.28. We can then use a modified form of the usual z-score formula to solve for X given a particular value of z.

• We can then use a modified form of the usual z-score formula to solve for X given a particular value of z.

• z= (x- µ )/δ X = µ+ zδ

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5

6.1

6

Population (or universe) – a group of a set of individuals that share at least one characteristic

Sample – a small number of individuals from the population

Social researchers generally are not able to measure an entire population

• Limited by time and resources

Sampling allows researchers to generalize

Populations and Samples

Population

sample

Sampling 6.2

Random Sampling

Nonrandom Sampling

vs.

Every member of the population

has the same chance of being

included

Every member of the population

does not have the same chance

of being included

Nonrandom Sampling 6.2

Accidental or Convenience

The sample is based on what is convenient for the researcher

Judgment or Purposive

The sample is drawn according to logic, common sense, or judgment

Random Sampling 6.2

Simple Random

Similar to drawing numbers from a hat

Systematic

Every nth member is included

Stratified

Divides the population into homogenous subgroups and then samples

6.3

10

By chance alone, we can always expect some difference between a sample and the population from which it is drawn

We use different symbols for samples as compared to populations

Sampling Error

Measure Population Sample

Mean

Standard Deviation

Standard Error

X

s

X

Xs

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• If you think about it realistically, it makes little sense that we would know the standard deviation of our variable in the population (s), but not know and need to estimate the population mean (µ).

• Indeed, there are very few cases when the population standard deviation is known.

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• The sampling distribution of means is no longer quite normal if we do not know the population standard deviation.

• In particular, the sampling distribution of means is a bit wider than a normal distribution.

• The ratio follows what is known as the t distribution, and thus it is called the t ratio.

13

• There is actually a whole family of t distributions.

• A concept known as degrees of freedom is used to determine which of the t distributions applies in a particular instance.

• The degrees of freedom indicate how close the t distribution comes to approximating the normal curve.

• When estimating a population mean, the degrees of freedom are one less than the sample size; that is,

• df = N - 1

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confidence intervals

© 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 • All Rights Reserved

6.5

Figure 6.6

17

1X

ss

N

estimated standard error of the mean

standard deviation of the sample

sample size

Xs

s

N

6.5

18

The range of mean values within which the population mean is likely to fall

• Confidence intervals can be calculated at different levels:

Confidence Intervals

68%CI 1.00X

X

95%CI 1.96X

X

99%CI 2.58X

X

• When dealing with the t distribution, we use T Table rather than Z Table.

• Unlike Z Table, for which we had to search out values of z corresponding to 95% and 99% areas under the curve, T Table is calibrated for special areas.

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• More precisely, T Table is calibrated for various levels of a (the Greek letter alpha).

• The a value represents the area in the tails of the t distribution. Thus, the a value is 1 minus the level of confidence. That is,

• a = 1 - level of confidence

• For example, for a 95% level of confidence, a = .05. For a 99% level of confidence, a = .01

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We enter T Table with two pieces of information:

(1) the degrees of freedom (N - 1)

(2) the alpha value, the area in the tails of the distribution.

For example, if we wanted to construct a 95% confidence interval with a sample size of 20,we would have 19 degrees of freedom (df = 20 - 1 = 19),

a = .05 and, as a result, a t value from T Table of 2.093.

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