The z-test

10.1Assume that a treatment does have an effect and that the treatment effect is being

evaluated with a z hypothesis test.If all factors are held constant, how is the outcome of the

hypothesis test influenced bysample size?To answer this question, do the following two tests

and compare the results.For both tests, a sample is selected from a normal population

distribution with a mean of = 32 and a standard deviation of = 9.After the treatment is

administered to the individuals in the sample, the sample mean is found to be M = 35.In each

case, use a two- tailed test with = .05.

a.For the first test, assume that the sample consists of n = 16 individuals.

b.For the second test, assume that the sample consists of n = 64 individuals.

c.Explain how the outcome of the hypothesis test is influenced by the

sample size.

(Note: Be sure and show a full diagram of the research design. Also show all steps and calculations

you made for each test following the process outlined in the z-test formula sheet handout. What

statistical decision do you make in each case?Finally report your results professionally in APA

format (see last step of the formula sheet for this module).

10.2Researchers at a National Weather Center in the northeastern United States recorded the

number of 90 degree days each year since records first started in 1875. The numbers form a normal

shaped distribution with a mean of = 9 and a standard deviation of = 2.1. To see if the data

showed any evidence of global warming, they also computed the mean number of 90 degree days for the

most recent n = 4 years and obtained M = 11.7 days. Do the data indicate that the past four years

have had significantly more 90 degree days than would be expected for a random sample from this

populaton? Use a one-tailed test with alpha = .05.

(Note: Be sure and show a diagram of the research design.

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Single Sample z-test

I.Assumptions for z-test

A.one sample, randomly selected

B.know population mean and population standard deviation ahead of time

C.standard deviation is unchanged by treatment or experiment

D.sample means are normally distributed; take all the possible sample means that could

happen by chance without treatment (usually normally distributed for behavioral sciences if sample

is greater than or equal to 30)

II.Diagramming your research (shows the whole logic and process of hypothesis testing)

a.Draw a picture of your research design (see diagramming your research handout).

b.There are always two explanations (i.e. hypotheses) of your research results, the

wording of which depends on whether the research question is directional (one- tailed) or

non-directional (two-tailed).State them as logical opposites.

c.For statistical testing, ignore the alternative hypothesis and focus on the null

hypothesis, since the null hypothesis claims that the research results happened by chance through

sampling error.

d.Assuming that the null is true (i.e. that the research results occurred by chance

through sampling error) allows one to do probability calculation (i.e. all statistical tests are

nothing more than calculating the probability of getting your research results by chance through

sampling error).

e.Observe that there are two outcomes which may occur from the results of the probability

calculation (high or low probability of getting your research results by chance, depending on the

alpha () level).

f.Each outcome will lead to a decision about the null hypothesis, whether the null is

probably true (i.e. we then accept the null to be true) or probably not true (i.e. we then reject

the null as false).

III.Hypotheses (i.e. the two explanations of your research results)

A.Two-tailed (non-directional research question)

1.Alternative hypothesis (H1): The independent variable (i.e. the treatment) does make a

difference in performance.

2.Null hypothesis (H0): The independent variable (i.e. the treatment) does not make a

difference in performance.

B.One-tailed (directional research question)

1.Alternative hypothesis (H1): The treatment has an increased (right tail) or a decreased

(left tail) effect on performance.

2.Null hypothesis (H0): The treatment has an opposite effect than expected or no change

in performance.

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IV.Determine critical regions (i.e. the z score boundary between the high or low

probability of getting your research results by chance) using table A-23

A.Significance level (should be given or decided prior to the research; also called the

confidence, alpha, or p level)

1. or p = .05, .01, or .001

B.One- or two-tailed test (using table A-23)

1.One-tailed: use full alpha level amount for proportion in tail (Column C)

2.Two-tailed: use half alpha level amount for proportion in tail (Column C)

C.With one- or two-tailed p values, find the critical z value

1.If two-tailed, then critical z value is z value

2.If one-tailed, then determine if critical z value is +z (right tail) or -z (left tail)

V.Calculate the z-test statistic

A.General Single Sample z-test statistical test formula

z =the observed sample mean - the hypothesized population mean

standard error

B.Calculations

1.Compute standard error (average difference between sample & population means)

Note: (standard error is simply an estimate of the average sampling error which may occur by

chance, since a sample can never give a totally accurate picture of a population)