Statistical Applications? Discussion Topic Hypothesis – A guess or prediction that states that something is true For this week…simply share a hypothesis. It can be about the class, the wo

Chapter 8 HYPOTHESIS TEST

In statistics, a hypothesis is a claim or statement about a property of a population.

A hypothesis test (or test of significance) is a standard procedure for testing a claim about a property of a population.

Examples of Hypotheses that can be Tested

• Genetics: The Genetics & IVF Institute claims that its XSORT method allows couples to increase the probability of having a baby girl.

• Business: A newspaper headline makes the claim that most workers get their jobs through networking.

• Medicine: Medical researchers claim that when people with colds are treated with echinacea, the treatment has no effect.

Examples of Hypotheses that can be Tested

• Aircraft Safety: The Federal Aviation Administration claims that the mean weight of an airline passenger (including carry-on baggage) is greater than 185 lb, which it was 20 years ago.

• Quality Control: When new equipment is used to manufacture aircraft altimeters, the new altimeters are better because the variation in the errors is reduced so that the readings are more consistent. (In many industries, the quality of goods and services can often be improved by reducing variation.)

Components of a Formal Hypothesis

Test

Null Hypothesis:

• The null hypothesis (denoted by ) is a statement that the value of a population parameter (such as proportion, mean, or standard deviation) is equal to some claimed value.

0H

0H

Alternative Hypothesis:

1H

1 a AH or H or H

Null and Alternative Hypotheses; Hypothesis Test

Null hypothesis: A hypothesis to be tested. We use the symbol H0 to represent the null hypothesis.

Alternative hypothesis: A hypothesis to be considered as an alternative to the null hypothesis. We use the symbol Ha to represent the alternative hypothesis.

Hypothesis test: The problem in a hypothesis test is to decide whether the null hypothesis should be rejected in favor of the alternative hypothesis.

Example:

Consider the claim that the mean weight of airline passengers (including carry-on baggage) is differ 195 lb (the current value used by the Federal Aviation Administration). Follow the two-steps procedure outlined in to identify the null hypothesis and the alternative hypothesis.

Example:

Step 2:

The test statistic is a value used in making a decision about the null hypothesis, and is found by converting the sample statistic to a score with the assumption that the null hypothesis is true.

Test Statistic

Test Statistic – Formulas

Test statistic for mean

or x x

z t s

n n

 



   

Significance Level

The significance level (denoted by ) is the probability that the test statistic will fall in the critical region when the null hypothesis is actually true. Common choices for are 0.05, 0.01, and 0.10.





Critical Value

A critical value is any value that separates the critical region (where we reject the null hypothesis) from the values of the test statistic that do not lead to rejection of the null hypothesis. The critical values depend on the nature of the null hypothesis, the sampling distribution that applies, and the significance level . See the previous figure where the critical value of z = 1.645 corresponds to a significance level of .



0.05 

Two-tailed Test

Means less than or greater than

 is divided equally between the two tails of the critical

region

0 :H 

1 :H 

Conclusions in Hypothesis Testing

We always test the null hypothesis. The initial conclusion will always be one of the following:

a. Reject the null hypothesis.

b. Fail to reject the null hypothesis.

Traditional method:

If the test statistic falls within the critical region, reject .

If the test statistic does not fall within the critical region, fail to reject .

Decision Criterion

0H

0H

Decision Criterion

Confidence Intervals:

A confidence interval estimate of a population parameter contains the likely values of that parameter. If a confidence interval does not include a claimed value of a population parameter, reject that claim.

Chapter 9 Learning Outcomes

• Know when to use t statistic instead of z-score hypothesis test

11

• Perform hypothesis test with t-statistics

22

Tools You Will Need

• Sample standard deviation (Chapter 4)

• Standard error (Chapter 7)

• Hypothesis testing (Chapter 8)

9.1 The t statistic: An alternative to z

• Sample mean (M) estimates (& approximates) population mean (μ)

• Standard error describes how much difference is reasonable to expect between M and μ.

• either or

n M

  

nM

2  

• Use z-score statistic to quantify inferences about the population.

• Use unit normal table to find the critical region if z-scores form a normal distribution – When n ≥ 30 or

– When the original distribution is approximately normally distributed

Reviewing the z-Score Statistic





and Mbetween distance standard

hypothesis and databetween difference obtained 

 

M

M z

The Problem with z-Scores

• The z-score requires more information than researchers typically have available

• Requires knowledge of the population standard deviation σ

• Researchers usually have only the sample data available

Introducing the t Statistic

• t statistic is an alternative to z.

• t might be considered an “approximate” z.

• Estimated standard error (sM) is used as in

place of the real standard error when the value of σM is unknown.

Estimated standard error

• Use s2 to estimate σ2.

• Estimated standard error:

• Estimated standard error is used as estimate of the real standard error when the value of σM is

unknown.

n

s or error standard

2

n

s sestimated M 

The t-Statistic

• The t statistic uses the estimated standard error in place of σM.

• The t statistic is used to test hypotheses about an unknown population mean μ when the value of σ is also unknown.

Ms

M t

 

Degrees of Freedom

• Computation of sample variance requires computation of the sample mean first – Only n-1 scores in a sample are independent

– Researchers call n-1 the degrees of freedom

• Degrees of freedom – Noted as df

– df = n-1

Figure 9.1 Distributions of the t Statistic

The t Distribution

• Family of distributions, one for each value of degrees of freedom

• Approximates the shape of the normal distribution – Flatter than the normal distribution

– More spread out than the normal distribution

– More variability (“fatter tails”) in t distribution

• Use Table of Values of t in place of the Unit Normal Table for hypothesis tests

Figure 9.2 The t Distribution for df=3

9.2 Hypothesis Tests with the t Statistic

• The one-sample t test statistic (assuming the null hypothesis is true)

0 error standard estimated

mean population -mean sample 

 

Ms

M t



Figure 9.3 Basic Research Situation for t Statistic

Hypothesis Testing: Four Steps

• State the null and alternative hypotheses and select an alpha level

• Locate the critical region using the t distribution table and df

• Calculate the t test statistic

• Make a decision regarding H0 (null hypothesis)

Figure 9.4 Critical Region in the t

Distribution for α = .05 and df = 8

Assumptions of the t Test

• Values in the sample are independent observations.

• The population sampled must be normal – With large samples, this assumption can be

violated without affecting the validity of the hypothesis test

Influence of Sample Size and Sample Variance

• The larger the sample, the smaller the error.

• The larger the variance, the larger the error.

Learning Check 1 (slide 1 of 4)

• When n is small (less than 30), the t distribution ______.

• is almost identical in shape to the normal z distributionAA

• is flatter and more spread out than the normal z distributionBB

• is taller and narrower than the normal z distributionCC

• cannot be specified, making hypothesis tests impossibleDD

Learning Check 1 – Answer (slide 2 of 4)

• When n is small (less than 30), the t distribution ______.

• is almost identical in shape to the normal z distributionAA

• is flatter and more spread out than the normal z distributionBB

• is taller and narrower than the normal z distributionCC

• cannot be specified, making hypothesis tests impossibleDD

Learning Check 1 (slide 3 of 4)

• Decide if each of the following statements is True or False.

• By chance, two samples selected from the same population have the same size (n = 36) and the same mean (M = 83). That means they will also have the same t statistic.

T/ F T/ F

• Compared to a z-score, a hypothesis test with a t statistic requires less information about the population.

T/ F T/ F

Learning Check 1 – Answers (slide 4 of 4)

• The two t values are unlikely to be the same; variance estimates (s2) differ between samples.

Fals e

Fals e

• The t statistic does not require the population standard deviation; the z-test does.

TrueTrue

9.3 Measuring Effect Size

• Hypothesis test determines whether the treatment effect is greater than chance – No measure of the size of the effect is included

– A very small treatment effect can be statistically significant

• Therefore, results from a hypothesis test should be accompanied by a measure of effect size.

Estimated Cohen’s d

• Original equation included population parameters

• Estimated Cohen’s d is computed using the sample standard deviation

s

M

deviation standardsample

difference mean d estimated

 

Figure 9.5 Distribution for Examples 9.1 & 9.2

Percentage of Variance Explained

• Determining the amount of variability in scores explained by the treatment effect is an alternative method for measuring effect size.

• r2 = 0.01 small effect

• r2 = 0.09 medium effect

• r2 = 0.25 large effect

dft

t

yvariabilit total

for accountedy variabilit r

 

2

2 2

Figure 9.6 Deviations with and without the Treatment Effect

Confidence Intervals for Estimating μ (slide 1 of 3)

• Alternative technique for describing effect size

• Estimates μ from the sample mean (M)

• Based on the reasonable assumption that M should be “near” μ

• The interval constructed defines “near” based on the estimated standard error of the mean (sM)

• Can confidently estimate that μ should be located in the interval

Confidence Intervals for Estimating μ (slide 2 of 3)

• Every sample mean has a corresponding t:

• Rearrange the equations solving for μ:

Ms

M t

 

MtsM 

Confidence Intervals for Estimating μ (slide 3 of 3)

• In any t distribution, values pile up around t = 0.

• For any α we know that (1 – α ) proportion of t values fall between ± t for the appropriate df.

• E.g., with df = 9, 90% of t values fall between ±1.833 (from the t distribution table, α = .10).

• Therefore, we can be 90% confident that a sample mean corresponds to a t in this interval.

Figure 9.7 t Distribution with df = 8

Confidence Intervals for Estimating μ (continued)

• For any sample mean M with sM

• Pick the appropriate degree of confidence (80%? 90%? 95%? 99%?) 90%

• Use the t distribution table to find the value of t (For df = 9 and α = .10, t = 1.833)

• Solve the rearranged equation

• μ = M ± 1.833(sM)

• Resulting interval is centered around M

• 90% confident that μ falls within this interval

Factors Affecting Width of Confidence Interval

• Confidence level desired

• More confidence desired increases interval width

• Less confidence acceptable decreases interval width

• Sample size • Larger sample smaller SE smaller interval

• Smaller sample larger SE larger interval

In the Literature

• Report whether (or not) the test was “significant”

• “Significant”  H0 rejected

• “Not significant”  failed to reject H0

• Report the t statistic value including df, e.g., t(12) = 3.65

• Report significance level, either: • p < alpha, e.g., p < .05 or

• Exact probability, e.g., p = .023

9.4 Directional Hypotheses and One-Tailed Tests

• Non-directional (two-tailed) test is most commonly used

• However, directional test may be used for particular research situations

• Four steps of hypothesis test are carried out – The critical region is defined in just one tail of the t

distribution.

Figure 9.8 Example 9.6 One-Tailed Critical Region

Learning Check 2 (slide 1 of 4)

• The results of a hypothesis test are reported as follows: t(21) = 2.38, p < .05. What was the statistical decision and how big was the sample?

• The null hypothesis was rejected using a sample of n = 21AA

• The null hypothesis was rejected using a sample of n = 22BB

• The null hypothesis was not rejected using a sample of n = 21CC

• The null hypothesis was not rejected using a sample of n = 22DD

Learning Check 2 – Answer (slide 2 of 4)

• The results of a hypothesis test are reported as follows: t(21) = 2.38, p < .05. What was the statistical decision and how big was the sample?

• The null hypothesis was rejected using a sample of n = 21AA

• The null hypothesis was rejected using a sample of n = 22BB

• The null hypothesis was not rejected using a sample of n = 21CC

• The null hypothesis was not rejected using a sample of n = 22DD

Learning Check 2 (slide 3 of 4)

• Decide if each of the following statements is True or False

• Sample size has a great influence on measures of effect size.

T/ F T/ F

• When the value of the t statistic is near 0, the null hypothesis should be rejected.

T/ F T/ F

Learning Check 2 – Answers (slide 4 of 4)

• Measures of effect size are not influenced to any great extent by sample size.

Fals e

Fals e

• When the value of t is near 0, the difference between M and μ is also near 0.

Fals e

Fals e