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Statistical Significance Testing: Older Students' Study Habits and Attitudes, Exams of Statistics

The concept of hypothesis testing and significance levels through an example of testing if older students have better study habits and attitudes towards school using the survey of study habits and attitudes (ssha) test. It covers the main concepts of hypothesis testing, null and alternative hypotheses, p-value, and guidelines for interpreting p-values.

Typology: Exams

Pre 2010

Uploaded on 08/19/2009

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TESTS OF SIGNIFICANCE – MOTIVATING EXAMPLE
The Survey of Study Habits and Attitudes (SSHA) is a psychological test that measures
students’ study habits and attitude toward school. Scores range from 0 to 200. The mean
score for U.S. college students is about 115. A teacher suspects that older students have
better attitudes toward school. She gives the SSHA to an SRS of 35 students who are at
least 30 years old. The sample results are = 125.7 and
s
= 30.1. Is this good evidence
that older students, on average, have better study habits and attitudes toward school than
the typical college student?
THE MAIN CONCEPTS OF HYPOTHESIS TESTING
A statistical test begins by supposing for the sake of argument that the effect we seek is
not present. We then look for evidence against this supposition and in favor of the effect
we hope to find.
For the null hypothesis, Ho, state a claim that we will try to find evidence against.
The null hypothesis is often a statement of "no effect" or "no difference". Nothing
special has occurred, no change has taken place -- the "status quo" hypothesis.
The statement we hope or suspect is true instead of Ho is the alternative
hypothesis, Ha.
A significance test looks for evidence against the null hypothesis and in favor of the
alternative hypothesis. The evidence is strong if the outcome we observe would rarely
come up when the null hypothesis is true.
That is, if the sample results can easily occur when Ho is true, we attribute the relatively
small discrepancy between the null hypothesis and the sample results to chance.
If the sample results cannot easily occur when Ho is true, we explain the relatively large
discrepancy between the null hypothesis and the sample results by concluding that Ho is
not true (and so we conclude that Ha is true).
Hints: (1) The null hypothesis will always contain equality.
(2) It's often easier to write down the alternative hypothesis first.
(3) P-value helps us assess the amount of evidence the sample provides against
Ho and in favor of Ha. P-value tells us how unlikely the sample results are
when Ho is true. Very small p-values mean the sample results are very
unlikely to occur when Ho is true and therefore the evidence against Ho is
strong.
(4) Language: Based on the p-value, we either "reject Ho in favor of H1" or we
"fail to reject Ho." (Sometimes I say “retain Ho” instead of “fail to reject Ho.)
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TESTS OF SIGNIFICANCE – MOTIVATING EXAMPLE

The Survey of Study Habits and Attitudes (SSHA) is a psychological test that measures

students’ study habits and attitude toward school. Scores range from 0 to 200. The mean

score for U.S. college students is about 115. A teacher suspects that older students have

better attitudes toward school. She gives the SSHA to an SRS of 35 students who are at

least 30 years old. The sample results are = 125.7 ands = 30.1. Is this good evidence

that older students, on average, have better study habits and attitudes toward school than

the typical college student?

THE MAIN CONCEPTS OF HYPOTHESIS TESTING

A statistical test begins by supposing for the sake of argument that the effect we seek is

not present. We then look for evidence against this supposition and in favor of the effect

we hope to find.

  • For the null hypothesis, H (^) o , state a claim that we will try to find evidence against.

The null hypothesis is often a statement of "no effect" or "no difference". Nothing

special has occurred, no change has taken place -- the "status quo" hypothesis.

  • The statement we hope or suspect is true instead of Ho is the alternative

hypothesis, H a.

A significance test looks for evidence against the null hypothesis and in favor of the

alternative hypothesis. The evidence is strong if the outcome we observe would rarely

come up when the null hypothesis is true.

That is, if the sample results can easily occur when Ho is true, we attribute the relatively

small discrepancy between the null hypothesis and the sample results to chance.

If the sample results cannot easily occur when Ho is true, we explain the relatively large

discrepancy between the null hypothesis and the sample results by concluding that Ho is

not true (and so we conclude that Ha is true).

Hints: (1) The null hypothesis will always contain equality.

(2) It's often easier to write down the alternative hypothesis first.

(3) P-value helps us assess the amount of evidence the sample provides against

H o and in favor of H a. P-value tells us how unlikely the sample results are

when H o is true. Very small p-values mean the sample results are very

unlikely to occur when Ho is true and therefore the evidence against H o is

strong.

(4) Language: Based on the p-value, we either "reject Ho in favor of H 1 " or we

"fail to reject H o ." (Sometimes I say “retain Ho ” instead of “fail to reject Ho .)

Guidelines for p-value

P-value is defined as the probability of obtaining sample results as extreme (or more

extreme) as those actually obtained, if Ho were true. (“Extreme” means far from what we

would expect if Ho were true. The alternative hypothesis determines which directions

count against H o .)

For example, p-value = .02 means sample results like those obtained only occur 2% of the

time when Ho is true.

P-value helps us assess the amount of evidence the sample provides against Ho and in favor

of H a. P-value tells us how unlikely the sample results are when Ho is true. Very small p-

values mean the sample results are very unlikely to occur when Ho is true and therefore

the evidence against H o is strong.

Language: Based on the p-value, we either "reject Ho in favor of Ha" or we "fail to reject

H o ." (Sometimes I say “retain Ho ” instead of “fail to reject Ho .)

The smaller the p-value, the stronger is the evidence against Ho. The following can be used

as guidelines when a significance level is not preset. They should not be viewed as p-value

“cutoffs.”

p-value > .1 insufficient evidence against Ho

.05 < p-value # .10 some evidence against H o

.01 < p-value # .05 fairly strong evidence against H o

.001 < p-value # .01 strong evidence against H o

p-value # .001 very strong evidence against Ho

Reporting a test of significance

1. Give the null and alternative hypotheses. Define the parameters involved in the

study.

2. Summarize the sample data for your readers.

3. Give the test statistic and its distribution, the observed test statistic, and the p-

value.

4. Use the p-value to draw a conclusion – reject the null hypothesis in favor of the

alternative or retain the null hypothesis. State your conclusion in context of the

problem.

  1. National Paper Company must purchase a new machine for producing cardboard boxes. The company must choose between two machines. Since the machines produce boxes of equal quality, the company will choose the machine that produces the most boxes in a one-hour period. The company selected eight assembly workers to test the two types of machines. The table below gives the number of boxes produced in an hour on each type of machine for each of these eight workers. Based on the data, is there sufficient evidence that the mean number of boxes produced in an hour differs for the two machines?

If there is a significant difference, estimate the difference with a 98% confidence interval. Interpret your interval estimate.

Machine Operator 1 2 3 4 5 6 7 8

Machine 1 53 60 58 48 46 54 62 49

Machine 2 50 55 56 44 45 50 57 47

  1. Ordinary corn doesn’t have as much of the amino acid lysine as animals need in their feed. Plant scientists have developed varieties of corn that have increased amounts of lysine. In a test of the quality of high-lysine corn as animal feed, an experimental group of 20 one-day- old chicks at a ration containing the new corn. A control group of another 20 chicks received a ration that was identical except that it contained normal corn. The weight gains after 21 days are given below.

Perform a hypothesis test to decide if high lysine corn is effective in increasing weight gain.

If you find that the high lysine corn does yield higher weight gains, estimate the mean extra weight gain with a 95% confidence interval.

Control

(normal corn)

Experimental

(high lysine corn)

  1. A production engineer is investigating whether there is a difference in the washer diameters manufactured by two different methods. A random sample of washers from the production line that uses the first method yields the following diameters (in inches).

0.861 0.864 0.882 0.887 0.858 0.879 0.887 0.876 0. 0.894 0.884 0.882 0.869 0.859 0.887 0.875 0.863 0. 0.882 0.862 0.906 0.880 0.877 0.864 0.873 0.860 0. 0.869 0.877 0.863 0.875 0.883 0.872 0.879 0.

The second method produces washers with the following diameters (in inches).

0.705 0.703 0.715 0.711 0.690 0.720 0.702 0.686 0. 0.712 0.718 0.695 0.708 0.695 0.699 0.715 0.691 0. 0.680 0.703 0.697 0.694 0.714 0.694 0.672 0.688 0. 0.715 0.709 0.698 0.696 0.700 0.706 0.695 0.

Do the data show a significant difference in the average diameters for the two methods? If yes, estimate the difference with a 95% confidence interval. Interpret your interval estimate.

  1. Advertisements for an instructional video claim that the techniques will improve the ability of Little League pitchers to throw strikes. To investigate this claim, we have 20 Little Leaguers throw 50 pitches each, and we record the number of strikes. After the players participate in the training program, we repeat the test. The table shows the number of strikes each player threw before and after the training.

Perform a hypothesis test to decide if the instructional video seems effective. If the instructional video is effective, use a 95% confidence interval to estimate the level of improvement in the ability of Little League pitchers to throw strikes.

Number of strikes (out of 50)

Number of Strikes (out of 50)

Before After Before After

28 35 33 33

29 36 33 35

30 32 34 32

32 28 34 30

32 30 34 33

32 31 35 34

32 32 36 37

32 34 36 33

32 35 37 35

33 36 37 32