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What factor determines whether you should use a z-test or a t-test statistic for a hypothesis test?

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PSYC 2317

Assignment #11

Single Sample t-Test

11.1 What factor determines whether you should use a z-test or a t-test statistic for a hypothesis test?

11.211.311.4A sample is selected from a population mean of µ = 30. A treatment is administered to the individuals in the sample and, after treatment, the sample mean found to be M = 31.3 with a sample variance of s2 = 12.

a. If the sample consists of n = 16 individuals, are the data sufficient to conclude that there is a significant treatment effect using a two-tailed test with α = .05?

b. If the sample consists of n = 36 individuals, are the data sufficient to conclude that there is a significant treatment effect using a two-tailed test with α = .05?

Ackerman & Goldsmith (2011) found that students who studied text from printed hard copy had better test scores than students who studied text presented on a screen. In a related study, a professor noticed that several students in a large class had purchased the e-book version of the course textbook. For the final exam, the overall average for the entire class was µ = 85, but the n = 10 students who purchased the e-book had a mean of M = 77 with a standard deviation of s = 7. Do these data indicate a significant decrease in test scores on the final exam due to the use of a e-book? Use a one-tailed test with α = .05.

Oishi & Shigehiro (2010) report that people who move from home to home frequently as children tend to have lower than average levels of well-being as adults. To further examine this relationship, a psychologist obtains a sample of n = 12 young adults who each experienced 5 or more different homes before they were 16 years old. These participants were given a standardized well-being questionnaire for which the general population has an average score of µ = 40. The well-being scores for this sample are as follows: 23, 37, 41, 35, 43, 37, 33, 34, 36, 38, 31, 38. On the basis of this sample, is well-being for frequent movers significantly different from the well-being of the general population? Test with alpha α = .05. Is this a onetailed or two-tailed test?

Note for problems 11.2-11.4 above: 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 t-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).

Single Sample t-test

I. Assumptions for t-test

A. Populations

1. the population from which the sample is selected is normal

B. One random sample (with replacement)

C. Data values

1. Sample values known (mean, standard deviation)

2. Population values (mean, standard deviation) not known

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 a 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 (α) 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.

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

1

confidence, alpha, or p level)

1. α or p = .05, .01, or .001

B. One- or two-tailed test

1. One-tailed: use the first row across the top

2. Two-tailed: use the second row across the top

C. Degrees of freedom

1. df = n – 1

D. With degrees of freedom & one- or two-tailed α value, find the critical t value

1. If two-tailed, then critical t value is ± t value

2. If one-tailed, then determine if critical t value is +t (right tail; expecting an increase) or –t (left tail; expecting a decrease)

V. Calculate t-test statistic

A. General statistical test formula

t = observed sample mean – hypothesized populational mean

standard error

B. Calculations

1. Compute variance

s or 

2. Compute standard error (average distance between sample & pop 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

sM = 

3. Compute t-test statistic (i.e. calculates the probability of getting your research results by chance through sampling error)

t = 

C. Compare the calculated t-score to the critical t-score & make a decision about the null hypothesis

1. Reject the null (as false) and accept the alternative or

2. Accept null (as true)

VI. Reporting the results of a Single Sample t test

“The treatment had a significant effect on (M = 25, SD = 4.22); t(18) = +3.00, p < .05, two-tailed.”

2

Single Sample t test Diagram Examplenon-treatment

t

reatment

Population

µ =

?

σ

=

unknown

high

Step 3: Using Part IV of formula sheet, determine the

Step 1: Diagram your study critical value(s)

(noting the data you are given)

n =

?

M = ?

s = ?

SS = ?

2 explanations of your data

(using directional or non-directional language depending on the research question)

H1: IV had an effect

single sample 2 outcomes 2 decisions t test High Prob.

(

easy)

Low Prob.

α

= .05

Prob.

calcul

ation

Accept H0

H0: IV had no effect (all results are (due to chance) due to chance primarily through

population Step 2: Create a table of

IV before tx calculations for each of the terms sample after which you will need to solve the tx

formulas (not needed in this problem)

x x2(data for sample)__ __∑= ∑=∑(x)2=The independent variable is being manipulated with two treatments

(hint: with an abstract problem like this, sometimes it helps to put in a real treatment, such as IV Pain Relievers with tx Ibuprofen and tx non-treatment) sampling error)

Reject H0 &

(hard) Accept H1

(due to real effect)

Step 4: Calculate the t statistic

1. s2 = SS Step 5: Compare critical t to

df the calculated t & make a

decision about the H0 s2 hypothesis

2. sm = n

Step 6: Report results

𝑀𝑀 – µ professionally (see last step of

3. t = formula sheet)

sm
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