Graded Exercises

6.6 2010 Healthcare Law.

On June 28, 2012 the U.S. Supreme Court upheld the much debated 2010 healthcare law, declaring it constitutional. A Gallup poll released the day after this decision indicates that 46% of 1,012 Americans agree with this decision. At a 95% con???dence level, this sample has a 3% margin of error. Based on this information, determine if the following statements are true or false, and explain your reasoning.

  1. We are 95% con???dent that between 43% and 49% of Americans in this sample support the decision of the U.S. Supreme Court on the 2010 healthcare law.

False, confidence intervals attempt to capture population proportion, not for the sample

  1. We are 95% con???dent that between 43% and 49% of Americans support the decision of the U.S. Supreme Court on the 2010 healthcare law.

True, confidence interval representing the population proportion

  1. If we considered many random samples of 1,012 Americans, and we calculated the sample proportions of those who support the decision of the U.S. Supreme Court, 95% of those sample proportions will be between 43% and 49%.

True, given the following conditions hold true: independent observations and success-failure proportion are greater than 10

  1. The margin of error at a 90% con???dence level would be higher than 3%.

False, it will be lower as ME is directly proportional to the Z* (critical value); 1.65 (at 90%) vs 1.96 (at 95%)

6.28 Sleep deprivation, CA vs. OR, Part I.

According to a report on sleep deprivation by the Centers for Disease Control and Prevention, the proportion of California residents who reported insufficient rest or sleep during each of the preceding 30 days is 8.0%, while this proportion is 8.8% for Oregon residents. These data are based on simple random samples of 11,545 California and 4,691 Oregon residents. Calculate a 95% con???dence interval for the difference between the proportions of Californians and Oregonians who are sleep deprived and interpret it in context of the data

pC - pO = 0.08 - 0.088 = -0.008 (point estimate of the difference of proportions)

SE_pC-pO = sqrt(pC(1-pC)/nC + pO(1-pO)/nO) = sqrt((0.08(1-0.08)/11545) + (0.088(1-0.088)/4691)) = 0.0048

At 95% confidence interval, -0.008 -/+ (1.96*0.0048) = (-0.0174, 0.0014) #### We are 95% confident that the difference between the proportions of Californians and Oregonians who are sleep deprived changes between -1.74% and 0.14%

6.44 Barking deer.

Microhabitat factors associated with forage and bed sites of barking deer in Hainan Island, China were examined from 2001 to 2002. In this region woods make up 4.8% of the land, cultivated grass plot makes up 14.7% and deciduous forests makes up 39.6%. Of the 426 sites where the deer forage, 4 were categorized as woods, 16 as cultivated grassplot, and 61 as deciduous forests. The table below summarizes these data

Woods Cultivated grassplot Deciduous forests Other Total 4 16 67 345 426

  1. Write the hypotheses for testing if barking deer prefer to forage in certain habitats over others.

H0: Barking deer do not prefer to forage in certain habitats over others; HA: Barking deer prefer to forage in certain habitats over others

  1. What type of test can we use to answer this research question?

Chi-square test - Given a sample of cases that can be classi???ed into several groups, determine if the sample is representative of the general population

  1. Check if the assumptions and conditions required for this test are satis???ed.

Independent counts; Expected counts should be greater or equal than 5: 4.8, 14,7, 39.6. All are good except for the Woods count, but we assume it is still fine to apply the chi-square distribution

  1. Do these data provide convincing evidence that barking deer prefer to forage in certain habitats over others? Conduct an appropriate hypothesis test to answer this research question

X^2 = (O1 - E1)^2/E1 + (Ok - Ek)^2/Ek

x^2 = ((((4/426)-4.8)^2)/4.8) + ((((16/426)-14.7)^2)/14.7) + ((((67/426)-39.6)^2)/39.6) + ((((345/426)-40.9)^2)/40.9) = 97.98

p_Value = 0, therefore, H0 should be rejected, Barking deer prefer to forage in certain habitats over others