identify the most appropriate test to use for the ...cathy/math2311/lectures/notes_nov28.pdf ·...

11/18/2018 Print Test

https://assessment.casa.uh.edu/Assessment/PrintTest.htm 2/5

Identify the most appropriate test to use for the following situation: In a experiment on relaxation techniques, subject's brain signals were measured before and after the

relaxation exercises. We wish to determine if the relaxation exercise slowed the brain waves.

a) Matched pairs

b) One sample t test

c) Two sample t test

d) Two sample z test

Question 5

To use the two sample t procedure to perform a significance test on the difference of two means, we assume:

a) The populations' standard deviation are known.

b) The samples from each population are independent.

c) The distributions are exactly normal in each population.

d) The sample sizes are large.

Question 6

Solid fats are more likely to raise blood cholesterol levels than liquid fats. Suppose a nutritionist analyzedthe percentage of saturated fat for a sample of 6 brands of stick margarine (solid fat) and for a sample of 6brands of liquid margarine and obtained the following results:

We want to determine if there a significant difference in the average amount of saturated fat in solid andliquid fats. What is the test statistic? (assume the population data is normally distributed)

a) t = 25.263

b) z = 39.604

c) t = 39.604

d) t = 39.104

e) z = 39.104

Question 7

It has been observed that some persons who suffer acute heartburn, again suffer acute heartburn within oneyear of the first episode. This is due, in part, to damage from the first episode. The performance of a newdrug designed to prevent a second episode is to be tested for its effectiveness in preventing a second episode.In order to do this two groups of people suffering a first episode are selected. There are 55 people in the firstgroup and this group will be administered the new drug. There are 75 people in the second group and this



group will be administered a placebo. After one year, 10% of the first group has a second episode and 9% ofthe second group has a second episode. Conduct a hypothesis test to determine, at the significance level 0.01,whether there is reason to believe that the true percentage of those in the first group who suffer a secondepisode is different from the true percentage of those in the second group who suffer a second episode?Select the [Alternative Hypothesis, Value of the Test Statistic].

a) [p1 = p2 , 0.1928]

b) [p1 ≠ p2 , 0.1928]

c) [p1 ≠ p2 , 0.2928]

d) [p1 < p2 , 0.1928]

e) [p1 > p2 , 0.1928]

f) None of the above

Question 8

It has been observed that some persons who suffer renal failure, again suffer renal failure within one year ofthe first episode. This is due, in part, to damage from the first episode. The performance of a new drugdesigned to prevent a second episode is to be tested for its effectiveness in preventing a second episode. Inorder to do this two groups of people suffering a first episode are selected. There are 31 people in the firstgroup and this group will be administered the new drug. There are 45 people in the second group and thisgroup will be administered a placebo. After one year, 11% of the first group has a second episode and 9% ofthe second group has a second episode. Conduct a hypothesis test to determine, at the significance level 0.1,whether there is reason to believe that the true percentage of those in the first group who suffer a secondepisode is more than the true percentage of those in the second group who suffer a second episode? Selectthe [Rejection Region, Decision to Reject (RH0) or Failure to Reject (FRH0)].

a) [z > 1.28, FRH0]

b) [z > -1.28 and z < 1.28, RH0]

c) [z < -1.28 and z > 1.28, FRH0]

d) [z < -1.28, FRH0]

e) [z < -1.28 or z > 1.28, RH0]


Question 9

A private and a public university are located in the same city. For the private university, 1048 alumni weresurveyed and 647 said that they attended at least one class reunion. For the public university, 797 out of 1319sampled alumni claimed they have attended at least one class reunion. Is the difference in the sampleproportions statistically significant? (Use α=0.05)



a) Reject the null hypothesis which states there is no difference in the proportion of alumni that attendedat least one class reunion in favor of the alternate which states there is a difference in the proportions.

b) Fail to reject the null hypothesis. There is not enough evidence to conclude that there is a difference inthe proportions.

c) There is not enough information to make a conclusion.

Question 10

Mars Inc. claims that they produce M&Ms with the following distributions:

Brown 30% Red 20% Yellow 20% Orange 10% Green 10% Blue 10%

A bag of M&Ms was randomly selected from the grocery store shelf, and the color counts were:

Brown 25 Red 24 Yellow 21 Orange 13 Green 17 Blue 13

Using the χ2 goodness of fit test to determine if the proportion of M&Ms is what is claimed, what is the teststatistic?

a) χ2 = 5.923

b) χ2 = 5.423

c) χ2 = 1.960

d) χ2 = 11.847

e) χ2 = 7.123


Question 11

Mars Inc. claims that they produce M&Ms with the following distributions:

Brown 30% Red 20% Yellow 20% Orange 10% Green 10% Blue 10%

A bag of M&Ms was randomly selected from the grocery store shelf, and the color counts were:

Brown 23 Red 22 Yellow 21 Orange 12 Green 16 Blue 13



Using the χ2 goodness of fit test (α = 0.10) to determine if the proportion of M&Ms is what is claimed.Select the [p-value, Decision to Reject (RH0) or Failure to Reject (FRH0)].

a) [p-value = 0.318, RH0]

b) [p-value = 0.682, RH0]

c) [p-value = 0.318, FRH0]

d) [p-value = 0.159, RH0]

e) [p-value = 0.682, FRH0]




PRINTABLE VERSIONQuiz 14

Question 1

State the type of hypothesis test to be used in the following situation:

Hippocrates magazine states that 37 percent of all Americans take multiple vitamins regularly. Suppose aresearcher surveyed 750 people to test this claim and found that 290 did regularly take a multiple vitamin. Isthis sufficient evidence to conclude that the actual percentage is different from 37%?

a) One Sample Z Test for Proportions

b) χ2 Goodness of Fit Test

c) Two Sample T Test for Means

d) χ2 Test for Independence

e) Matched Pairs T Test

f) One Sample T Test for Means

g) Two Sample Z Test for Proportions

h) One Sample Z Test for Means

Question 2


Solid fats are more likely to raise blood cholesterol levels than liquid fats. Suppose a nutritionist analyzedthe percentage of saturated fat for a sample of 6 brands of stick margarine (solid fat) and for a sample of 6brands of liquid margarine and obtained the following results:

Stick 25.8 26.9 26.2 25.3 26.7 26.1 Liquid 16.9 17.4 16.8 16.2 17.3 16.8

Is there a significant difference in the average amount of saturated fat in solid and liquid fats? Assume thepopulation is normally distributed.

a) Two Sample T Test for Means

b) χ2 Goodness of Fit Test

c) Two Sample Z Test for Proportions



d) χ2 Test for Independence

e) Matched Pairs T Test

f) One Sample T Test for Means

g) One Sample Z Test for Proportions

h) One Sample Z Test for Means

Question 3


In a certain city, there are about one million eligible voters. A simple random sample of size 10,000 waschosen to study the relationship between gender and participation in the last election. The results were:

Men Women Voted 2792 3591 Didn't Vote 1486 2131

Does there appear to be a relationship between gender and participation in the last election?

a) Two Sample Z Test for Proportions

b) One Sample Z Test for Means

c) χ2 Test for Independence

d) One Sample Z Test for Proportions

e) One Sample T Test for Means

f) χ2 Goodness of Fit Test

g) Two Sample T Test for Means

h) Matched Pairs T Test

Question 4



If we are testing for a relationship between gender and participation in the last election, what is the test



statistic?

a) z = -20.564

b) χ2 = 17.578

c) χ2 = 26.367

d) z = -17.505

e) χ2 = 15.789

Question 5



If we are testing for a relationship between gender and participation in the last election, what is the p-valueand decision at the 5% significance level? Select the [p-value, Decision to Reject (RH0) or Failure to Reject(FRH0)]

a) [p-value = 0.687, RH0]

b) [p-value = 0.224, RH0]

c) [p-value = 0.224, FRH0]

d) [p-value = 0.112, RH0]

e) [p-value = 0.687, FRH0]

Question 6

The Blue Diamond Company advertises that their nut mix contains (by weight) 40% cashews, 15% Brazilnuts, 20% almonds and only 25% peanuts. The truth-in-advertising investigators took a random sample (ofsize 20 lbs) of the nut mix and found the distribution to be as follows: 5 lbs of Cashews, 5 lbs of Brazil nuts,8 lbs of Almonds and 2 lbs of Peanuts. At the 0.05 level of significance, is the claim made by Blue Diamondtrue?

Select the [p-value, Decision to Reject (RH0) or Failure to Reject (FRH0)].

a) [p-value = 0.004, RH0]



b) [p-value = 0.020, RH0]

c) [p-value = 0.041, RH0]

d) [p-value = 0.004, FRH0]

e) [p-value = 0.041, FRH0]

Question 7

Quart cartons of milk should contain at least 32 ounces. A sample of 20 cartons contained the followingamounts in ounces. Does sufficient evidence exist to conclude the mean amount of milk in cartons is lessthan 32 ounces at the 5% significance level?

The data is: (32.5, 32.8, 31.8, 27.7, 32.2, 31.7, 31.8, 31.3, 28.6, 31.7, 27.5, 28.9, 32.3, 32.5, 27.8, 31.2, 31.1,32.6, 27.7, 27.8)


a) [p-value = 0.997, RH0]

b) [p-value = 0.003, RH0]

c) [p-value = 0.003, FRH0]

d) [p-value = 0.997, FRH0]

e) [p-value = 0.001, RH0]

Question 8

Hippocrates magazine states that 34 percent of all Americans take multiple vitamins regularly. Suppose aresearcher surveyed 750 people to test this claim and found that 262 did regularly take a multiple vitamin. Isthis sufficient evidence to conclude that the actual percentage is different from 34% at the 5% significancelevel?


a) [p-value = 0.295, RH0]

b) [p-value = 0.589, FRH0]

c) [p-value = 0.639, FRH0]

d) [p-value = 0.589, RH0]

e) [p-value = 0.295, FRH0]

Question 9



A national computer retailer believes that the average sales are greater for salespersons with a collegedegree. A random sample of 36 salespersons with a degree had an average weekly sale of $3588 last year,while 32 salespersons without a college degree averaged $3322 in weekly sales. The standard deviationswere $468 and $642 respectively. Is there evidence at the 5% level to support the retailer's belief?


a) [p-value = 0.015, FRH0]

b) [p-value = 0.031, FRH0]

c) [p-value = 0.031, RH0]

d) [p-value = 0.016, RH0]

e) [p-value = 0.015, RH0]

Question 10

The community hospital is studying its distribution of patients. A random sample of 320 patients presently inthe hospital gave the following information:

Type of Patient Old Rate of Occurrences Present Number of Occurrences Maternity Ward 20% 72 Cardiac Ward 32% 93 Burn Ward 10% 30 Children's Ward 15% 51 All Other Wards 23% 74

Test the claim at the 5% significance level that the distribution of patients in these wards has not changed.


a) [p-value = 0.352, RH0]

b) [p-value = 0.703, RH0]

c) [p-value = 0.140, RH0]

d) [p-value = 0.703, FRH0]

e) [p-value = 0.140, FRH0]

Question 11

In a experiment on relaxation techniques, subject's brain signals were measured before and after therelaxation exercises with the following results:

Person 1 2 3 4 5



Before 37.8 37.7 42.4 37.9 42.4 After 35.9 39.7 37.5 39.9 40.9

Is there sufficient evidence to suggest that the relaxation exercise slowed the brain waves? Assume thepopulation is normally distributed.


a) [p-value = 0.547, RH0]

b) [p-value = 0.273, FRH0]

c) [p-value = 0.273, RH0]

d) [p-value = 0.547, FRH0]

e) [p-value = 0.137, RH0]

Exam 3 ReviewReview

Cathy Poliak, [email protected] in Fleming 11c

Department of MathematicsUniversity of Houston

Exam Review

Cathy Poliak, Ph.D. [email protected] Office in Fleming 11c (Department of Mathematics University of Houston )Review Exam Review 1 / 25

Outline

1 Material Covered

2 Confidence Intervals

3 Hypothesis Tests

4 What is on the exam


Popper Set Up

Fill in all of the proper bubbles.

Make sure your ID number is correct.

Make sure the filled in circles are very dark.

This is popper number 22.


Chapters Covered for Exam 3

Inference for parameters.Chapter 7 - Confidence Intervals

Chapter 8 - Hypothesis Tests


Assumptions for Inferences

To estimate the mean(s):1. The sample(s) is a simple random sample (SRS).2. The population is a Normal distribution or approximately Normal

(Central Limit Theorem).3. I The population standard deviation, σ is given - use z.

I The sample standard deviation, s is given - use t .

To estimate the proportion(s):1. The sample(s) is a simple random sample (SRS).2. The population size is ten times larger than the sample size.3. The number of success (np) and number of failures (n(1-p)) have

to be at least 10.


Confidence Intervals for One Parameter

The intervals are calculated depending on what you are given. Thefollowing table gives you a step by step approach:

Parameter µ given σ µ not given σ p proportions1. Estimate x̄ =

∑x

n x̄ =∑

xn p̂ = x

n2. Confidence Level C = 1− α this is given in the problem3. Critical value z∗ t∗ with df = n − 1 z∗

4. Standard error σ√n

s√n

√p̂(1−p̂)

n

5. Margin of error Critical value × Standard error6. Confidence interval Point estimate ± Margin of error


Confidence Intervals for the Difference of TwoParameters

The intervals are calculated depending on what you are given andwhat you want to find. The following table gives you a step by stepapproach:

Parameter µd µ1 − µ2 p1 − p2 proportions1. Estimate x̄d =

∑d

n x̄1 − x̄2 p̂1 − p̂2

2. Confidence Level C = 1 − α this is given in the problem3. Critical value t∗ with t∗ with df smaller z∗

df = n − 1 of n1 − 1 or n2 − 1

4. Standard error sd√n

√s2

1n1

+s2

2n2

√p̂1(1−p̂1)

n1+ p̂2(1−p̂2)

n2

5. Margin of error Critical value × Standard error6. Confidence interval Point estimate ± Margin of error


Behavior of Confidence Intervals with different C andwith different n

The width (length) of the interval is highest value minus lowest value or2 times the margin of error.

As the confidence level C decreases, the width (length) alsodecreases.

As the confidence level C increases, the width(length) alsoincreases.

As the sample size n decreases, the width (length) increases.

As the sample size n increases, the width (length) decreases.


Determining Sample Sizes

To determine the sample size for proportions use

n =

(z∗

m

)2

× p∗ × (1− p∗)

Where p∗ is a prior knowledge of proportion or p∗ = 0.5 if notgiven.

The confidence interval for a population mean will have aspecified margin of error m when the sample size is

n =

(z∗ × σ

m

)2

Where the sample size is the next whole number.


Types of Tests and Their Test Statistics

One Sample Z-test: Test for mean given the population standarddeviation, σ.

z =x̄ − µ0

σ/√

n

One Sample T-test: Test for mean given the sample standarddeviation, s.

t =x̄ − µ0

s/√

n

With df = n − 1.

One Sample Proportions test: Test for proportions (neither a meannor a standard deviation is given).

z =p̂ − p0√p0(1−p0)

n


Types of Tests and Their Test Statistics Continued

Matched Pairs Tests: Test for the mean difference, the samplesare dependent on each other.

t =x̄d − µd

sd/√

nWith df = n − 1.Two Sample T-test: Test for the difference of two means, thesamples are independent of each other.

t =x̄1 − x̄2√

s21

n1+

s22

n2

Where degress of freedom is the smaller of n1 − 1 or n2 − 1.Two Sample Proportions Tests: Test for the difference of twoproportions.

z =p̂1 − p̂2√

p1(1−p1)n1

+ p2(1−p2)n2


Types of Tests and Their Test Statistics Continued

χ2 Goodness of Fit Test: Test for one categorical variable.

χ2 =∑ (observed count− expected count)2

expected count

Where expected count = total number of counts × proportion ofeach category and df = number of categories− 1.

χ2 Test for Independence: Test for dependence between twocategorical variables.

χ2 =∑ (observed count− expected count)2

expected count

Where expected count = row total×column totaln and df = (r − 1)(c− 1).


Rejection Regions

The rejection region depends on:1. Alternative hypothesis.2. Level of significance: α = 1− c. This is given.3. Test statistic: z, t or χ2.

Test Ha : µ < µ0 Ha : µ > µ0 Ha : µ 6= µ0Z-test z < qnorm(α) z > qnorm(1− α) z < qnorm(α/2)

or z > qnorm(1− α/2)T-test t < qt(α,df) t > qt(1− α,df) t < qt(α/2,df)

or t > qt(1− α/2,df)χ2 test χ2 > qchisq(1− α,df )

If the test statistic is in the rejection region then we will reject the nullhypothesis. Otherwise, we fail to reject H0.


P-value

The P-value is the probability of getting the observed value (teststatistic) or extreme (in the way of the alterntative hypothesis)assuming the null hypothesis is true. The p-value depends on:

1. Alternative hypothesis.2. Test statistic.

Test Ha : µ < µ0 Ha : µ > µ0 Ha : µ 6= µ0Z-test pnorm(z) 1 - pnorm(z) 2∗pnorm(-z)T-test pt(t,df) 1 - pt(t,df) 2∗pt(-t,df)χ2 test 1 - pchisq(χ2,df)

If the p-value is less than (or equal) to α then we reject the nullhypothesis. Otherwise, we fail to reject H0.


Conclusion

The conclusion is going back and answering the question based on thetest if we reject or fail to reject H0. Put this in context of the problem.

Reject H0: We have significance, the alternative is correct.

Fail to reject H0: We do not have significance, the alternative isnot correct.


What to Expect on the Exam

The test is only multiple choice.1. 17 Multiple choice questions.

2. First 7 questions are worth 10 points each.

3. Last 10 questions are based on 2 hypothesis test, proportionsand/or means (one or two samples). 5 questions to eachhypothesis test. Each question worth 3 points.


Example of Multiple Choice Questions

A simple random sample of 100 8th graders at a large suburbanmiddle school indicated that 86% of them are involved with some typeof after school activity. Find the 98% confidence interval that estimatesthe proportion of them that are involved in an after school activity.a) (0.679, 0.891)b) (0.699, 0.941)c) (0.779, 0.941)d) (0.829, 0.834)e) (0.779, 0.741)


Example 2

An SRS of 24 students at UH gave an average height of 6.1 feet and astandard deviation of .3 feet. Construct a 90% confidence interval forthe mean height of students at UH.a) (4.600, 7.900)b) (6.079, 6.121)c) (5.995, 6.205)d) (5.586, 6.614)e) (4.850, 7.550)


Example 3

A 98% confidence interval for the mean of a population is to beconstructed and must be accurate to within 0.3 unit. A preliminarysample standard deviation is 1.7. The smallest sample size n thatprovides the desired accuracy isa) 183b) 180c) 174d) 185e) 164


Example 4

In a hypothesis test, if the computed P-value is less than 0.001, thereis very strong evidence toa) retest with a different sample.b) fail to reject the null hypothesis.c) reject the null hypothesis.


Example 5

What will reduce the width of a confidence interval?a) Increase variance.b) Increase confidence level.c) Decrease variance.d) Decrease number in sample.


Example 6

In a two-tailed hypothesis test situation HA : µ 6= µ0, the test statistic isdetermined to be t = −2.423. The sample size has been 41. Thep-value for this test isa) -0.01b) +0.01c) -0.02d) +0.02


Example 7

A six-sided die is thrown 50 times. The numbers of occurrences ofeach face are shown below.

Face 1 2 3 4 5 6Count 12 5 9 11 6 7

Can you conclude that the die is not fair? What type of test should beused in this situation and what is the test statistic?a) Two proportion z test; z = 2.5514b) One proportion z test; z = 1.176c) χ2 Goodness of Fit; χ2 = 2.956d) χ2 Goodness of Fit; χ2 = 4.72


What You Need an What is Provided

ProvidedI Basic calculator; it will be a link you see in the exam.I R; it will be a link you see in the exam.I z-table, T-table, χ2 table; it will be a link you see in the exam.I Formula sheet; it will be a link you see in the exam.

Can bringI Calculator; if it is memory based CASA will remove the memory.I Pencil; you will need something to write with for the free response

questions.I Your Cougar Card.


Questions?


Bags of a certain brand of tortilla chips claim to have a net weight of 14 ounces. Net weights actually vary slightly from bag to bag and are normally distributed with mean µ. A representative of a consumer advocate group wishes to see if there is any evidence that the mean net weight is less than advertised. Test this claim at the 5% significance level. 4. Determine the null hypothesis H0 and alternative hypothesis HA.

a) H0: µ = 14, HA: µ ≠ 14 b) H0: µ = 14, HA: µ < 14

c) H0: µ = 14, HA: µ > 14 d) H0: �̅�𝑥 = 14, HA: �̅�𝑥 < 14

5. To do this test, he selects 16 bags of this brand at random and determines the net weight of each. He finds

the sample mean to be 𝑥𝑥�̅�𝑥 = 13.8668 and the sample standard deviation to be s = 0.25. Calculate the test statistic for this significance test. a) t = -2.131 b) z = -2.131

c) t = -0.5328 d) t = -0.1332

6. Give the rejection region.

a) t < -1.753 b) z < -1.645

c) t < -2.131 or t > 2.131 d) z < -1.96 or z > 1.96

7. Determine the p-value.

a) P-value = 0.025 b) P-value = 0.05

c) P-value = 0.975 d) P-value = 0.95

8. State the conclusion.

a) Reject the null hypothesis; the mean net weight is significantly less than 14 ounces. b) Fail to reject the null hypothesis; the mean net weight is significantly less than 14 ounces. c) Reject the null hypothesis; the mean net weight is not significantly less than 14 ounces. d) Fail to reject the null hypothesis; the mean net weight is not significantly less than 14 ounces.

identify the most appropriate test to use for the ...cathy/math2311/lectures/notes_nov28.pdf ·...

Documents