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Unit 6 Starters

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Unit 6 Starters Starter 11.1.1 Draw a standard normal curve on your calculator Set your window to [-3,3]1 by [-.1,.4] .1 Use y1 = normalpdf(x) Draw a neat, reasonably ... – PowerPoint PPT presentation

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Title: Unit 6 Starters


1
Unit 6 Starters
2
Starter 11.1.1
  • Draw a standard normal curve on your calculator
  • Set your window to -3,31 by -.1,.4 .1
  • Use y1 normalpdf(x)
  • Draw a neat, reasonably large sketch of the curve
    in your notes. Show scales.
  • We will use this later

3
Starter 11.1.2
  • The heights of 25 students have a mean of 186
    cm with standard deviation 3 cm.
  • What is the standard error of the distribution of
    sample means?
  • Assuming the population mean height (µ) of the
    population is 185, what is the t test statistic?
  • What is the probability of getting a sample mean
    of 186 or higher?

4
Starter answers
  • SE 3/v25 3/5 .6
  • P-value from the table
  • Row 24 1.318 lt t lt 1.711
  • So .10 gt P gt .05
  • P-value from the calculator
  • tcdf(1.67,999,24)0.054

5
Starter 11.1.3
  • Scores on the AP exam are supposed to have a mean
    of 3. Mr. McPeak thinks his students score
    higher than that, so he takes a sample of ten of
    last years scores
  • Here are the scores
  • Using the methods we learned yesterday
  • Find a 95 confidence interval for the mean from
    the formula
  • Perform a hypothesis test that might support Mr.
    McPeaks claim
  • Clearly state your conclusion in a sentence

2 2 5 3 4
3 5 4 1 4
6
Starter answers
  • From 1-var Stats Ë 3.3 S 1.337
  • SE 1.337/sqrt(10) .4228
  • t 2.262
  • C.I. 3.3 0.956 (2.343, 4.257)
  • Ho µ 3 Ha µ gt 3 Choose a .05
  • Note Choice of a is arbitrary but reasonable.
  • t (3.3 3) / .4228 .709
  • P tcdf(.709,999,9) .248
  • Conclusion There is not sufficient evidence (t
    .709, P .248) to support the claim that the
    mean is greater than 3

7
Starter 11.1.4
  • Seven students took an SAT prep course after
    doing poorly on the test. Here are their before
    (row 1) and after (row 2) scores
  • Use a matched-pairs t-test on your calculator to
    determine if they improved their scores in a
    statistically significant way

1020 1100 1110 1000 990 1050 1060
1030 1120 1100 1050 1040 1110 1160
8
Starter solution
  • Put before in L1 and after in L2
  • Let L3 L2 L1 (or reverse)
  • State hypotheses
  • Ho µ 0
  • Ha µ gt 0
  • Stat Tests T-Test
  • Choose Data, µo 0, L3, µ gt µo
  • There is good evidence (t6 2.90, p .01) to
    support the claim that the SAT prep course led to
    improved scores.

9
Starter 11.1.5
  • Write the important elements that must be present
    in any experimental design? What is the
    desireable (but not mandatory) element of
    experimental design?
  • Comparison
  • There should be a treatment group and a control
    group
  • Randomization
  • Subjects must be randomly assigned to a group
  • Replication
  • There must be a large enough sample in each group
    that the results are statistically significant
  • Blindness (optional but highly desirable)
  • Subjects should not know which group they are in
  • Experiments should not know either (if possible)

10
Starter 11.2.1
  • Write the assumptions that underlie the use of
    the t test.
  • Which assumption is so important you cant work
    without it?
  • What is the best way to tell if the distribution
    assumption is met?

11
Starter Solution
  • The two main assumptions we need are
  • The sample came from a valid SRS of the
    population.
  • The population is approximately normally
    distributed.
  • We cant do anything without a valid SRS.
  • Plot the data to see if they are approximately
    normal.
  • Note that if sample size is large we can live
    with skewness or outliers.

12
Starter 11.2.2
  • In the early eighties, a large group of 13 year
    olds were given the SAT. Verbal results were
    nearly identical, but there was a real difference
    in the math results. Here are the facts
  • 19,883 boys had a mean score of 416 with standard
    deviation of 87
  • 19,937 girls had a mean score of 386 with a
    standard deviation of 74
  • Use the formula to state a 99 confidence
    interval for the mean difference of scores
    between ALL boys and girls.

13
Starter solution
  • From the table, t 2.576
  • So we are 99 confident that the true difference
    of the boys mean score minus the girls mean
    score is between 27.9 and 32.1

14
Starter 12.1.1
  • Do dogs who are house pets have higher
    cholesterol than dogs who live in a research
    clinic? A clinic measured the cholesterol level
    in all 23 of its dogs and found a mean level of
    174 with s.d. of 44. They also measured 26 house
    pets brought in to be neutered one week and found
    a mean of 193 with s.d. of 68.
  • Is this strong evidence that house pets have
    higher cholesterol than clinic dogs?
  • What is wrong with this study?

15
Solution
  • Use the calculators 2-sample TTest
  • t 1.174 p 0.123
  • There is not enough evidence to support the claim
    that house pets have higher cholesterol than
    clinic dogs.
  • The problem with this study is that there is not
    proper randomization. The clinic used all its
    dogs and used pets that happened to be in the
    clinic for treatment. This casts serious doubt
    on the validity of the conclusion.

16
Starter 12.1.2
  • Some people think that chemists are more likely
    than others to have female children. Perhaps
    they are exposed to chemicals that cause this.
    Between 1980 and 1990 in Washington state, 555
    children were born to chemists. Of these births,
    273 were girls. During this period, 48.8 of all
    births in Washington were girls. Is there
    evidence that the proportion of girls born to
    chemists is higher than normal?
  • Write hypotheses, calculate the sample
    proportion, perform a test, and write your
    conclusion.

17
Starter solution
  • Ho p .488 Ha p gt .488
  • p-hat 273 / 555 .492, so find z and p
  • p normalcdf(.188,999) .43
  • There is not good evidence (p .43) that the
    proportion of chemists girls is higher than
    normal.

18
Starter 12.2.1
  • One-sample procedures for proportions can also
    be used in matched pairs experiments. Here is an
    example
  • Each of 50 randomly selected subjects tastes
    two unmarked cups of coffee and says which he/she
    prefers. One cup in each pair contains instant
    coffee the other is fresh-brewed. 31 of the
    subjects prefer fresh-brewed.
  • Test the claim that a majority of people prefer
    the taste of fresh-brewed coffee. State
    hypotheses, check assumptions, find the test
    statistic and p-value. Is your result
    significant at the 5 level?
  • Find a 90 confidence interval for the true
    proportion that prefer fresh-brewed.
  • When you do an experiment like this, in what
    order should you present the two cups of coffee
    to the subjects?

19
Starter Solution
  • Ho p .5 Ha p gt .5
  • Assumptions
  • SRS? OK
  • Large population? All coffee drinkers gt 10x50
    OK
  • At least 10 in each group? 50 x .5 gt 10 OK
  • z 1.70 p .045
  • There is sufficient evidence (plt.05) to support
    the claim that coffee drinkers prefer
    fresh-brewed.
  • C.I. .62 1.645 x .0686 (.507, .733)
  • Randomization is needed in any experiment flip a
    coin (or use another method) to choose which
    coffee each subject gets first.
  • See question on next slide

20
The Cola Challenge!
  • Do Northgate students prefer Pepsi over Coke? In
    a randomized matched-pairs experiment we did last
    fall, here were the results
  • Preferred Pepsi 58
  • Preferred Coke 31
  • Perform a hypothesis test of the claim that Pepsi
    is preferred.

21
Starter 12.2.2
  • The drug AZT is used to treat symptoms of AIDS.
    It was studied in an experiment involving
    volunteers already diagnosed as having HIV, the
    virus that causes AIDS. 435 subjects took AZT
    and another 435 took a placebo. At the end of
    the study, 17 of the AZT subjects had developed
    AIDS 38 of the placebo subjects had developed
    AIDS. We want to test the claim that taking AZT
    lowers the proportion of people who go from HIV
    to AIDS.
  • Assign numbers to the groups
  • Verify that assumptions are met and state
    hypotheses
  • Carry out the test on calculator and write your
    conclusion
  • This experiment was double-blind. What does that
    mean?

22
Starter Solution
  • Let Group 1 be the AZT Group 2 the placebo
  • Assumptions
  • SRS
  • Each population at least 10 times sample size
  • Each count of yes or no at least 5
  • Ho p1 p2 Ha p1lt p2
  • StatTests2-PropZTest yields Z -2.93 and
    p .0017
  • Conclude there is strong evidence to support the
    claim that AZT reduces proportion who get AIDS
  • Double Blind Neither the subject nor the person
    administering the drug knows if the subject gets
    the AZT or the placebo

23
Starter 13.1.1
  • Sickle-cell trait is a hereditary condition that
    is common among blacks and can cause medical
    problems. Some biologists suggest that the trait
    protects against malaria. A study in Africa
    tested 543 children for the sickle-cell trait and
    also for malaria. In all, 136 of the children
    had the sickle-cell trait, and 36 of these had
    heavy malaria infections. The other 407 children
    lacked the trait, and 152 of them had heavy
    malaria infections.
  • What are the two populations of interest here?
  • Give a 95 Confidence Interval for the difference
    in proportions of malaria in the two populations.
  • Is there good evidence that the proportion of
    heavy malaria infection is lower among children
    with the sickle-cell trait?

24
Answer
  • The two populations are children with the trait
    and children without the trait.
  • Use the TI 2-PropZInt screen (-.197, -.021)
  • I am 95 confident that the malaria proportion in
    children with sickle-cell is between 2 and 20
    less than in children without the trait.
  • Because 0 was not in the confidence interval,
    there is good evidence to support the claim that
    the sickle-cell trait protects against malaria.
  • Note If you run the 2-PropZTest, z -2.3, p
    .01

25
Starter 13.1.2
  • Elite distance runners are thinner than the rest
    of us. Skinfold thickness, which indirectly
    measures body fat, can show this. A random
    sample of 20 runners had a mean skinfold of 7.1
    mm with a standard deviation of 1.0 mm. A random
    sample of 95 non-runners had a mean of 20.6 w/ sd
    of 9.0.
  • Form a 95 confidence interval for the mean
    difference in body fat between runners and
    non-runners.

26
Starter Solution
  • Choose 2-SampTInt
  • Enter x17.1 Sx11 n120
  • Enter x220.6 Sx29 n295
  • Enter C .95
  • Find (-15.38, -11.62)
  • Conclusion We are 95 confident that the true
    mean skinfold of runners is between 15.4 mm and
    11.6 mm less than non-runners.

27
Starter 13.1.3
  • According to the NCAA, 45 out of 74 athletes
    admitted to a certain university in 1994
    graduated within 6 years. Assuming this is a
    valid sample of all athletes, does the proportion
    of athletes who graduate differ from the
    all-university proportion, which is 68?
  • State hypotheses, perform a test, write a
    conclusion.

28
Starter Solution
  • This is a one-sample proportion test
  • Now what are the hypotheses?
  • Ho p .68 Ha p ? .68
  • Use StatTests1-PropZTest
  • po .68 x 45 n 74
  • z -1.32 p .185
  • Conclusion There is not sufficient evidence (p
    .185) to support a claim that the graduation
    rate of athletes differs from non-athletes.
  • Is there a different approach that could be taken
    to get the same result?

29
Starter 13.2.1
  • A study of iron deficiency in infants compared
    two groups. One had been breast-fed, the other
    had been fed formula from a bottle. The
    hemoglobin levels were measured at age 12 months.
    Here are the results
  • Assuming this was a properly randomized
    experiment, is there significant evidence that
    the mean hemoglobin level is different between
    the groups?
  • Why is the assumption that we had a properly
    designed experiment questionable?

Group n mean Std dev
Breast-fed 23 13.3 1.7
Bottle-fed 19 12.4 1.8
30
Starter Solution
  • This is a two-sample means problem
  • Assumptions needed
  • SRS from each population
  • Independent populations
  • Sample means approximately normally distributed
  • Use 2-SampTTest or 2-SampTInt
  • Ho µ1 µ2 Ha µ1 ? µ2 a .05
  • t 1.65 p .107
  • There is not sufficient evidence (p .107) to
    support a claim that there is a difference in the
    hemoglobin level between the two groups.

31
Starter 13.2.2
  • I rolled a die 60 times and got the following
    distribution of results
  • Is the die fair? Perform a test and state your
    conclusion.

Outcome 1 2 3 4 5 6
Quantity 6 10 7 11 8 18
32
Starter Solution
  • Observed outcomes in L1 6, 10, 7, 11, 8, 18
  • Expected outcomes in L2 all 10s
  • (O E)²/E in L3
  • Sum L3 to get X² of 9.4
  • X²cdf(9.4, 999, 5) .094
  • Conclusion There is not sufficient evidence
    (p.09) to support a claim that the die is unfair.

33
Dice Day Starter
  • Do unregulated providers of child care in their
    homes follow different health and safety
    practices in different cities? A study looked at
    people who regularly provided care for someone
    elses children in poor areas of three cities
    The numbers who required medical releases from
    parents to allow medical care in an emergency
    were 42 of 73 providers in Newark, 29 of 101 in
    Camden and 48 of 107 in Chicago.
  • Is there a significant difference among the
    proportions of providers who require medical
    releases in the three cities?
  • Identify the two variables and write a two-way
    table of counts.
  • Write the null and alternative hypotheses.
  • Perform the test and draw a conclusion.
  • Verify that the necessary conditions are met.

34
Solution
  • Ho There is no association between city and
    requirement.
  • Ha There is an association between city and
    requirement.
  • Run X2 test on calc.
  • There is good evidence to support a claim that
    requirements differ by city
  • Check expected counts in matrix B

Require Not Req.
Newark 42 31
Camden 29 72
Chicago 48 59
35
Starter 14.1.1
  • The Goodwill second-hand stores did a survey of
    their customers in Walnut Creek and Oakland.
    Among other things, they noted the sex of each
    respondent. Here is the breakdown
  • Is there a significant difference between the
    proportion of women customers in the two stores?
  • Treat this as a two-sample proportion problem
  • Find the z statistic and p value draw a
    conclusion
  • Do a chi-square test
  • Find X² and the p value draw a conclusion
  • How does X² relate to z?

Men Women
W.C. 38 203
Oakland 68 150
36
Starter Solution
  • Ho p1 p2
  • Ha p1 ? p2
  • Use the 2-PropZTest on the TI
  • Find z 3.92 And p 9.0 x 10-5
  • Conclude that there is strong evidence that the
    proportions differ
  • Use the chi-square test on the TI
  • Find X² 15.334 And p 9.0 x 10-5
  • Conclude that there is strong evidence that the
    proportions differ
  • Note that X² is the square of the z statistic
  • Conclusion A two-proportion test can be done
    with either the z statistic or with a chi-square
    test
  • The result is the same

37
Starter 14.1.2
  • The Goodwill Stores of Walnut Creek and Oakland
    also did a breakdown of their shoppers by income.
    Here are the results
  • Is there good evidence to believe that the
    customers of the two stores have different income
    distributions?

Income (1000s) W.C. Oakland
Under 10 70 62
10 20 52 63
20 25 69 50
25 35 22 19
35 28 24
38
Starter Solution
  • Ho There is no difference in the income
    distributions
  • Ha The income distributions differ
  • Put the two-way table into matrix A
  • Run the chi-square Test
  • X2 3.955 p .412
  • Conclusion There is not sufficient evidence (p
    .412) to support a claim that the income
    distributions are different

39
Starter 14.3.1
  • Men and women were observed playing a game of
    chance. 3 of 12 men won the game 8 of 12 women
    won.
  • Is there a statistically significant difference
    between the mens and womens results?
  • State hypotheses, perform a test and write a
    conclusion

40
Starter solution
  • This asks for a comparison of proportions from
    two populations
  • Use 2-PropZTest screen
  • Choose p1 ? p2 alternative
  • Get z 2.05 and p 0.04
  • Or use X2 Test screen
  • Get X2 4.196 and p .04
  • Conclusion
  • There is good evidence (p .04) that the winning
    proportions are different for men and women
  • Checking matrix B shows all expected counts are
    at least 5.
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