2003 Free Response Question

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2003 Free Response Question 2A simple random
sample of adults living in a suburb of a large
city was selected. The age and annual income of
each adult in the sample were recorded. The
resulting data are summarized in the table below.

• What is the probability that a person chosen at
  random from those in the sample will be in the
  31-45 age category?
• What is the probability that a person chosen at
  random from those in this sample whose incomes
  are over 50,000 will be in the 31-45 age
  category? Show your work.
• Based on your answers to parts (a) and (b), is
  annual income independent of age category for
  those in this sample? Explain.

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12.1Tests for the Mean of a Population
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T-Statistic

• There is a different t-distribution for each
  sample size
• We specify a particular t distribution by giving
  its degrees of freedom n-1.
• Notation t(k) t-distribution with k degrees of
  freedom.

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Facts about t-distributions

• Density curve is similar in shape to standard
  normal curve.
• Spread is a bit greater than that of the standard
  normal dist.
• As d.o.f. k increase, the t(k) density curve
  approaches N(0,1) more closely.

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T-distribution critical values.

• Find the p-value from Table C for a
  t-distribution with n 20 and t 1.81. Ha mean
  gt 5.
• Find the p-value from Table C for a
  t-distribution with n 37 and t -3.17. Ha
  two-sided.

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One-sample T Procedures

• Draw an SRS of size n from a population having
  unknown mean. A level C confidence interval for
  is
• To test the hypothesis based on an SRS of
  size n, compute the one-sample t statistic

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Cola Problem

• Cola makers test new recipes for loss of
  sweetness during storage. Trained tasters
  selected from an SRS rate the sweetness before
  and after storage. Here are the sweetness losses
  found by 10 tasters for one new cola recipe
• 2.0 0.4 0.7 2.0 -0.4 2.2 -1.3 1.2 1.1
  2.3
• Are these good data evidence that the cola lost
  sweetness? (Sweetness loss sweetness
  before-sweetness after storage)

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Matched Pairs t-procedures

• Recall In a matched pairs design, subjects are
  matched in pairs and each treatment is given to
  one subject in each pair, or we can perform
  before-and-after observations on the same
  subjects.
• Taste-test example is a matched pairs study in
  which the same 10 tasters rated before-and-after
  sweetness.
• To compare the responses to the two treatments in
  a matched pairs design, apply the one-sample t
  procedures to the observed differences.
• The parameter in a matched pairs t procedure
  is the mean difference in the responses to the 2
  treatments within matched pairs of subjects in
  the entire population.

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Floral Scents and Learning

• We hear that listening to Mozart improves
  students performance on tests. Perhaps pleasant
  odors have a similar effect. To test this idea,
  21 subjects chosen from an SRS worked a
  pencil-and-paper maze while wearing a mask the
  mask was either unscented or had a floral scent.
  The response variable is their average time on 3
  trials. Each subjected worked the maze with both
  masks, in random order. The following table gives
  the subjects average times with both masks.

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• Analysis Subtract the scented time from the
  unscented time for each subject.
• The 21 differences form a single sample.
• First subject 7.37 s slower wearing the scented
  mask, so difference was -.
• Shorter times better performance, therefore
  differences student did better.

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• The developer of a new filter for filter-tipped
  cigarettes claims that it leaves less nicotine in
  the smoke than does the current filter. Because
  cigarette brands differ in a number of ways, he
  tests each filter on one cigarette of each of
  nine brands and records the difference in
  nicotine content (current filter new filter).
  The mean difference is x-bar 1.32 milligrams
  (mg), and the standard deviation of the
  differences is s 2.35 mg.
• 1.      What is the parameter being measured?
• 2.      State Ho and Ha for this study in both
  symbols and words.
• 3.      Describe the population of interest for
  which inference is being performed.
• 4.      What conditions are required to carry out
  the significance test? Discuss the validity of
  each.
• 5.      Determine the test statistic and the
  P-value. Show your work.
• 6.      What do you conclude?
• 7. Construct a 90 confidence interval for the
  mean amount of additional nicotine removed by the
  new filter is. Interpret the interval.