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% confidence 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% confident 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: we are 100% confident that 46% of SAMPLE respondents approved the decision.

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

True

  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

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

False:

Margin of error = Z * Standard of Error

Z score at 90% is lower than Z score at 95%, hence margin of error is less.

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 insuffient 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% confidence interval for the difference between the proportions of Californians and Oregonians who are sleep deprived and interpret it in context of the data.

Checked assumptions:

All assumptions are met.

ca_p <- 0.08
ca_n <- 11545
or_p <- 0.088
or_n <- 4691
SE <- sqrt((or_p*(1-or_p))/or_n + (ca_p*(1-ca_p))/ca_n)
Z<- 1.96
Diff <- or_p - ca_p
lb <- (Diff - Z*SE)
ub <- (Diff + Z*SE)
lb
## [1] -0.001498128
ub
## [1] 0.01749813

The 95% confidence interval for the difference in proportion of Oregon and California residents for sleep deprivation ranges from -0.0015 to 0.0175.

The confidence interval includes zero it means that there is NOT enough evidence to reject the H0 hypotesis and we can NOT conclude that there is a difference in sleep deprivation between California and Oregon residents.

6.44

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.

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

H0: barking deer prefer to forage same habitats as others

H1: barking deer prefer to forage different habitats over others

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

A chi-square test

  1. Check if the assumptions and conditions required for this test are satisfied.
wood <- 426 * 0.048
wood
## [1] 20.448
cg <- 426 * 0.147
cg
## [1] 62.622
df <- 426 * 0.396
df
## [1] 168.696
other <- 426 * (1-(0.048 + 0.147 + 0.396))
other
## [1] 174.234

All conditions are satisfied.

  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.
chi <- (4-20.448)^2/20.448 + (16-62.622)^2/62.622 + (61-168.696)^2/168.696 +(345-174.234)^2/174.234
chi
## [1] 284.0609
pchisq(chi, 3, lower.tail = FALSE)
## [1] 2.799724e-61

If p-value < 0.05, we have enough evidence to reject H0 hypothesis in favor of alternative one.