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10.7 Hypothesis testing of two means and two proportions: homework  (Page 4/4)

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Let the subscript t size 12{t} {} = treated patient and ut size 12{ ital "ut"} {} = untreated patient.

The appropriate hypotheses are:

  • H o : p t < p ut size 12{H rSub { size 8{o} } :p rSub { size 8{t} }<p rSub { size 8{ ital "ut"} } } {} and H a : p t ≥ p ut size 12{H rSub { size 8{a} } :p rSub { size 8{t} }>= p rSub { size 8{ ital "ut"} } } {}
  • H o : p t ≤ p ut size 12{H rSub { size 8{o} } :p rSub { size 8{t} }<= p rSub { size 8{ ital "ut"} } } {} and H a : p t > p ut size 12{H rSub { size 8{a} } :p rSub { size 8{t} }>p rSub { size 8{ ital "ut"} } } {}
  • H o : p t = p ut size 12{H rSub { size 8{o} } :p rSub { size 8{t} } =p rSub { size 8{ ital "ut"} } } {} and H a : p t ≠ p ut size 12{H rSub { size 8{a} } :p rSub { size 8{t} }<>p rSub { size 8{ ital "ut"} } } {}
  • H o : p t = p ut size 12{H rSub { size 8{o} } :p rSub { size 8{t} } =p rSub { size 8{ ital "ut"} } } {} and H a : p t < p ut size 12{H rSub { size 8{a} } :p rSub { size 8{t} }<p rSub { size 8{ ital "ut"} } } {}

D

If the p -value is 0.0062 what is the conclusion (use α = 0.05 size 12{α=5%} {} )?

  • The method has no effect.
  • There is sufficient evidence to conclude that the method reduces the proportion of HIV positive patients that develop AIDS after four years.
  • There is sufficient evidence to conclude that the method increases the proportion of HIV positive patients that develop AIDS after four years.
  • There is insufficient evidence to conclude that the method reduces the proportion of HIV positive patients that develop AIDS after four years.

B

Lesley E. Tan investigated the relationship between left-handedness and right-handedness and motor competence in preschool children. Random samples of 41 left-handers and 41 right-handers were given several tests of motor skills to determine if there is evidence of a difference between the children based on this experiment. The experiment produced the means and standard deviations shown below. Determine the appropriate test and best distribution to use for that test.

Left-handed Right-handed
Sample size 41 41
Sample mean 97.5 98.1
Sample standard deviation 17.5 19.2

  • Two independent means, normal distribution
  • Two independent means, student's-t distribution
  • Matched or paired samples, student's-t distribution
  • Two population proportions, normal distribution

B

For questions [link] – [link] , use the following information.

The Eastern and Western Major League Soccer conferences have a new Reserve Division that allows new players to develop their skills. Data for a randomly picked date showed the following annual goals.

Western Eastern
Los Angeles 9 D.C. United 9
FC Dallas 3 Chicago 8
Chivas USA 4 Columbus 7
Real Salt Lake 3 New England 6
Colorado 4 MetroStars 5
San Jose 4 Kansas City 3

Conduct a hypothesis test to determine if the Western Reserve Division teams score, on average, fewer goals than the Eastern Reserve Division teams. Subscripts: 1 Western Reserve Division ( W ); 2 Eastern Reserve Division ( E )

The exact distribution for the hypothesis test is:

  • The normal distribution.
  • The student's-t distribution.
  • The uniform distribution.
  • The exponential distribution.

B

If the level of significance is 0.05, the conclusion is:

  • There is sufficient evidence to conclude that the W Division teams score, on average, fewer goals than the E teams.
  • There is insufficient evidence to conclude that the W Division teams score, on average, more goals than the E teams.
  • There is insufficient evidence to conclude that the W teams score, on average, fewer goals than the E teams score.
  • Unable to determine.

C

Questions [link] – [link] refer to the following.

Neuroinvasive West Nile virus refers to a severe disease that affects a person’s nervous system . It is spread by the Culex species of mosquito. In the United States in 2010 there were 629 reported cases of neuroinvasive West Nile virus out of a total of 1021 reported cases and there were 486 neuroinvasive reported cases out of a total of 712 cases reported in 2011. Is the 2011 proportion of neuroinvasive West Nile virus cases more than the 2010 proportion of neuroinvasive West Nile virus cases? Using a 1% level of significance, conduct an appropriate hypothesis test. (Source: http:// http://www.cdc.gov/ncidod/dvbid/westnile/index.htm )

  • “2011” subscript: 2011 group.
  • “2010” subscript: 2010 group

This is:

  • a test of two proportions
  • a test of two independent means
  • a test of a single mean
  • a test of matched pairs.

A

An appropriate null hypothesis is:

  • p 2011 = p 2010 size 12{p rSub { size 8{ ital "drugless"} }<= p rSub { size 8{ ital "drug"} } } {}
  • p 2011 ≥ p 2010 size 12{P rSub { size 8{ ital "drugless"} }>= P rSub { size 8{ ital "drug"} } } {}
  • μ 2011 ≤ μ 2010 size 12{μ rSub { size 8{ ital "drugless"} }>= μ rSub { size 8{ ital "drug"} } } {}
  • p 2011 > p 2010 size 12{P rSub { size 8{ ital "drugless"} }>P rSub { size 8{ ital "drug"} } } {}

A

The p size 12{p} {} -value is 0.0022. At a 1% level of significance, the appropriate conclusion is

  • There is sufficient evidence to conclude that the proportion of people in the United States in 2011 that got neuroinvasive West Nile disease is less than the proportion of people in the United States in 2010 that got neuroinvasive West Nile disease.
  • There is insufficient evidence to conclude that the proportion of people in the United States in 2011 that got neuroinvasive West Nile disease is more than the proportion of people in the United States in 2010 that got neuroinvasive West Nile disease.
  • There is insufficient evidence to conclude that the proportion of people in the United States in 2011 that got neuroinvasive West Nile disease is less than the proportion of people in the United States in 2010 that got neuroinvasive West Nile disease.
  • There is sufficient evidence to conclude that the proportion of people in the United States in 2011 that got neuroinvasive West Nile disease is more than the proportion of people in the United States in 2010 that got neuroinvasive West Nile disease.

D

Suppose a statistics instructor believes that there is no significant difference between the mean class scores of statistics day students on Exam 2 and statistics night students on Exam 2. She takes random samples from each of the populations. The mean and standard deviation for 35 statistics day students were 75.86 and 16.91. The mean and standard deviation for 37 statistics night students were 75.41 and 19.73. The “day” subscript refers to the statistics day students. The “night” subscript refers to the statistics night students.

An appropriate alternate hypothesis for the hypothesis test is:

  • μ day > μ night size 12{μ rSub { size 8{8:"30"} }>μ rSub { size 8{"11":"30"} } } {}
  • μ day < μ night size 12{μ rSub { size 8{8:"30"} }<μ rSub { size 8{"11":"30"} } } {}
  • μ day = μ night size 12{μ rSub { size 8{8:"30"} } =μ rSub { size 8{"11":"30"} } } {}
  • μ day ≠ μ night size 12{μ rSub { size 8{8:"30"} }<>μ rSub { size 8{"11":"30"} } } {}

D

A concluding statement is:

  • There is sufficient evidence to conclude that statistics night students mean on Exam 2 is better than the statistics day students mean on Exam 2.
  • There is insufficient evidence to conclude that the statistics day students mean on Exam 2 is better than the statistics night students mean on Exam 2.
  • There is insufficient evidence to conclude that there is a significant difference between the means of the statistics day students and night students on Exam 2.
  • There is sufficient evidence to conclude that there is a significant difference between the means of the statistics day students and night students on Exam 2.

C
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Source:  OpenStax, Collaborative statistics using spreadsheets. OpenStax CNX. Jan 05, 2016 Download for free at http://legacy.cnx.org/content/col11521/1.23
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