INFERENCES FROM TWO SAMPLES

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INFERENCES FROM TWO SAMPLES
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Overview
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Hypothesis Testing
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Hypothesis Testing

• State the claim (in words).
• State the null and alternative hypotheses.
• Obtain the test statistic.
• Determine One-tail/Two-tail test, obtain critical
  value(s) or p-Value(s).
• Reject/Fail to reject H0.
• State conclusion (in words).

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Inferences About Two Proportions
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Hypothesis Testing
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Requirements for Inferences About Two Proportions

• Requirements
• We have proportions from two simple random
  samples that are independent, which means that
  the sample values selected from one population
  are not related to or somehow paired or matched
  with the sample values selected from the other
  population.
• For each of the two samples, the number of
  successes is at least five, and the number of
  failures is at least five.
• Null Hypothesis H0 p1 p2

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Requirements for Inferences About Two Proportions
(continued)

• Notation for Two Proportions
• For population 1 we letp1 population
  proportionn1 size of the sample taken from
  population 1x1 number of successes in the
  sample from population 1
  (the sample proportion)The corresponding
  meanings are attached to p2, n2, x2, , and
  , which come from population 2.

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Pooled Estimate of p1 and p2

• The pooled estimate of p1 and p2 is denoted by
  and is given byWe denote the compliment of
  by , so

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Test Statistic for Two Proportions (with H0 p1
p2)

• where p1 p2 0
  and

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Testing Claims About a Population Proportion P
(continued)

• P-values Use Table A-2 or calculator.
• Critical values Use Table A-2.

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Testing Claims About a Population Proportion P
(calculator)

• 2-PropZTestx1 is the number of successes in
  group 1n1 is the number of trials (sample size)
  for group 1x2 is the number of successes in
  group 2n2 is the number of trials (sample size)
  for group 2

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Hypothesis Testing

• State the claim (in words).
• State the null and alternative hypotheses.
• Obtain the test statistic.
• Determine One-tail/Two-tail test, obtain critical
  value(s) or p-Value(s).
• Reject/Fail to reject H0.
• State conclusion (in words).

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Example

• Gender differences in student needs and fears
  were examined in the article A Survey of
  Counseling Needs of Male and Female College
  Students (J. College Student Development
  (1998)). Random samples of male and female
  students were selected from those attending a
  particular university. Of 234 males surveyed, 64
  said that they were concerned about the
  possibility of getting AIDS. Of 568 female
  students surveyed, 243 reported being concerned
  about the possibility of getting AIDS. Is there
  sufficient evidence to conclude that the
  proportion of female students concerned about the
  possibility of getting AIDS is greater than the
  corresponding proportion of males? Test at the
  level of significance.

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Confidence Interval Estimate of p1 p2

• The confidence interval estimate of the
  difference p1 p2 iswhere

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Confidence Interval Estimate of p1 p2
(calculator)

• 2-PropZIntx1 is the number of successes in group
  1n1 is the number of trials (sample size) for
  group 1x2 is the number of successes in group
  2n2 is the number of trials (sample size) for
  group 2

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Example

• Use the sample data from the previous example to
  construct a 95 confidence interval for the
  difference between the two population proportions.

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Inferences About Two Means Independent Samples
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Hypothesis Testing
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Definitions

• Two samples are independent if the sample values
  selected from one population are not related to
  or somehow paired or matched with the sample
  values selected from the other population.
• Two samples are dependent if the members of one
  sample can be used to determine the members of
  the other samples. Samples consisting of matched
  pairs are dependent.

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Testing Claims About Two Population Means
Independent Samples

• Requirements
• The two samples are independent.
• Both samples are simple random samples.
• Either or both of these conditions is satisfied
• The two sample sizes are both large (with
  and ), or
• both samples come from populations having normal
  distributions.
• Null Hypothesis

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Testing Claims About Two Population Means
Independent Samples

• Hypothesis Test Statistic for Two Means
  Independent Samples

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Testing Claims About Two Population Means
Independent Samples

• Degrees of Freedom When finding critical values
  or P-values, us the following for determining the
  number of degrees of freedom, denoted by df.
• To obtain a simple, conservative estimatedf
  smaller of n1 1 and n2 1.
• To obtain a more accurate dfwhere
  and

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Testing Claims About Two Population Means
Independent Samples

• P-values Refer to Table A-3 or calculator.
• Critical values Refer to the t values in Table
  A-3.

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Testing Claims About Two Population Means
Independent Samples (calculator)

• 2-SampTTest (using summary statistics) is
  the sample mean for group 1Sx1 is the sample
  standard deviation for group 1 n1 is the sample
  size for group 1 is the sample mean for
  group 2Sx2 is the sample standard deviation for
  group 2 n2 is the sample size for group 2

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Testing Claims About Two Population Means
Independent Samples (calculator)

• 2-SampTTest (using data) List1 is the name of
  list containing the data for group 1List2 is the
  name of list containing the data for group
  2Freq1 is the frequency of entries in List1,
  (should be 1)Freq2 is the frequency of entries
  in List2, (should be 1)Pooled can we assume the
  standard deviations are equal?
  (default is no)

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Hypothesis Testing

• State the claim (in words).
• State the null and alternative hypotheses.
• Obtain the test statistic.
• Determine One-tail/Two-tail test, obtain critical
  value(s) or p-Value(s).
• Reject/Fail to reject H0.
• State conclusion (in words).

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Example

• The discharge of industrial wastewater into
  rivers affects water quality. To assess the
  effect of a particular power plant on water
  quality, 24 water specimens were take 16 km
  upstream and 4 km downstream of the plant.
  Alkalinity (mg/L) was determined for each
  specimen, resulting in the summary quantities
  given below. Does the data suggest that there is
  a difference in the mean alkalinity of the river?
  Assume that the two samples are selected from
  populations that are approximately normally
  distributed. Test at the 0.05 level of
  significance.

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Confidence Interval Estimate of
Independent Samples

• The confidence interval estimate of the
  difference iswhereand the
  number of degrees of freedom (df) is as described
  above for hypothesis tests.

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Testing Claims About Two Population Means
Independent Samples

• and are known.Test Statistic
• Confidence intervalwhere

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Testing Claims About Two Population Means
Independent Samples

• Requirements
• The two populations have the same standard
  deviation. That is, .
• The two samples are independent.
• Both samples are simple random samples.
• Either or both of these conditions is satisfied
• The two sample sizes are both large (with
  and ), or
• both samples come from populations having normal
  distributions.

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Testing Claims About Two Population Means
Independent Samples

• Hypothesis Test Statistic for Two Means
  Independent Samples and
  whereand the number of degrees of
  freedom is given bydf n1 n2 2.

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Confidence Interval Estimate of
Independent Samples and

• Confidence intervalwhereand is as
  given in the above test statistic and the number
  of degrees of freedom is given by df n1 n2
  2.

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Inferences for Matched Pairs
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Hypothesis Testing
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Testing Claims About Two Population Means
Matched Pairs

• Requirements
• The sample data consists of matched pairs.
• The samples are simple random samples.
• Either or both of these conditions is satisfied
• the number of matched pairs of sample data is
  large ( ), or
• the pairs of values have differences that are
  from a distribution that is approximately normal.
• Null Hypothesis

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Testing Claims About Two Population Means
Matched Pairs

• Notation for Matched Pairsd individual
  difference between the two values in a
  single matched pair mean value of the
  differences d for the population of all
  matched pairs mean value of the differences
  d for the paired sample data standard
  deviation of the differences d for the
  paired sample datan number of pairs

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Testing Claims About Two Population Means
Matched Pairs

• Hypothesis Test Statistic for Matched Pairs
  where degrees of freedom n 1
• P-values and Critical values Table A-3 (t
  distribution with n 1 degrees of freedom) or
  calculator

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Hypothesis Testing

• State the claim (in words).
• State the null and alternative hypotheses.
• Obtain the test statistic.
• Determine One-tail/Two-tail test, obtain critical
  value(s) or p-Value(s).
• Reject/Fail to reject H0.
• State conclusion (in words).

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Example

• The effect of exercise on the amount of lactic
  acid in the blood was examined in the article A
  Descriptive Analysis of Elite-Level Racquetball
  (Research Quarterly for Exercise and Sport
  (1991)). Eight males were selected at random from
  those attending a week-long training camp. Blood
  lactate levels were measured before and after
  playing three games of racquetball. The results
  are given in the table below. Can we conclude
  that the mean amount of lactic acid in the blood
  after exercise is higher than the mean amount of
  lactic acid in the blood before exercise? Test at
  the 0.05 level of significance.

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Confidence Interval for Matched Pairs

• where

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Comparing Variation in Two Samples
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Hypothesis Testing
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Testing Claims About Two Population Variances or
Standard Deviations

• Methods of Variations standard deviation of
  the samples2 variance of sample standard
  deviation of the population variance of
  population

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The F Distribution

• The F distribution is not symmetric.
• Values of the F distribution cannot be negative.
• The exact shape of the F distribution depends on
  two different degrees of freedom.

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Testing Claims About Two Population Variances or
Standard Deviations

• Requirements
• The populations are independent of each other.
• The populations are each normally distributed.
• Notation for Hypothesis Tests with Two Variances
  or Standard Deviations
• larger of the two sample variances size
  of the sample with the larger variance variance
  of the population from which the sample with
  the larger variance is drawn
• The symbols , , and are used
  for the other sample and population

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Testing Claims About Two Population Variances or
Standard Deviations

• Test Statistic for Hypothesis Tests with Two
  Variances (where is the larger of the
  two sample variances)

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Testing Claims About Two Population Variances or
Standard Deviations

• P-values and Critical values Use calculator or
  Table A-5 to find critical F values that are
  determined by the following
• The significance level .
• Numerator degrees of freedom n1 1
• Denominator degrees of freedom n2 1

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Testing Claims About Two Population Variances or
Standard Deviations (calculator)

• 2-SampFTest (using data) List1 is the name of
  list containing the data for group 1List2 is the
  name of list containing the data for group
  2Freq1 is the frequency of entries in List1,
  (should be 1)Freq2 is the frequency of entries
  in List2, (should be 1)

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Testing Claims About Two Population Variances or
Standard Deviations (calculator)

• 2-SampFTest (using summary statistics) Sx1 is
  the sample standard deviation for group 1 n1 is
  the sample size for group 1Sx2 is the sample
  standard deviation for group 2 n2 is the sample
  size for group 2

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Hypothesis Testing

• State the claim (in words).
• State the null and alternative hypotheses.
• Obtain the test statistic.
• Determine One-tail/Two-tail test, obtain critical
  value(s) or p-Value(s).
• Reject/Fail to reject H0.
• State conclusion (in words).

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Example

• In a study published in the Archives of General
  Psychiatry entitled Efficacy of Olanzapine in
  Acute Bipolar Mania, researchers conducted a
  randomized, doubleblind study to measure the
  effects of the drug olanzapine on patients
  diagnosed with bipolar disorder. One hundred
  fifteen patients with a DSM-IV diagnosis of
  bipolar disorder were randomly divided into two
  groups, a treatment group that received 5 to 20
  mg per day of olanzapine, and a control group
  that received a placebo. The effectiveness of
  the drug was measured by the Young-Mania rating
  Scale total score, with the net improvement in
  the score recorded. The results are given in the
  table below. Assuming that the data are
  approximately normally distributed, test the
  claim that the standard deviation in the
  treatment group is different from the standard
  deviation in the control group at the 0.05 level
  of significance.