Chapter 6 - Inference for Categorical Data

2010 Healthcare Law. (6.48, p. 248) 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.

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% certain of the results released by the Gallup poll. No need for a confidence interval.

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 with a 3% margin of error, the poll result toward 43% and 49% of Americans supporting the decision with a 95% confidence interval.

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, sample proportion will be within the confidence interval.

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

=> False the margin of error will be lower the confidence interval is smaller.

Legalization of marijuana, Part I. (6.10, p. 216) The 2010 General Social Survey asked 1,259 US residents: “Do you think the use of marijuana should be made legal, or not” 48% of the respondents said it should be made legal.

Is 48% a sample statistic or a population parameter? Explain.

=> 48% is a sample statistic as this isn’t the entire USA Population.

Construct a 95% confidence interval for the proportion of US residents who think marijuana should be made legal, and interpret it in the context of the data.

n <- 1259
p <- 0.48
cl <- 0.95

se <- (((p) * (1-p)) / (n)) ^ (0.5)
se

## [1] 0.01408022

ubound <- 0.48 + 1.96 * se
lbound <- 0.48 - 1.96 * se
c(lbound, ubound)

## [1] 0.4524028 0.5075972

=> lower and upper bounds are 0.4524028 0.5075972

A critic points out that this 95% confidence interval is only accurate if the statistic follows a normal distribution, or if the normal model is a good approximation. Is this true for these data? Explain.

ph <- n*p
ph

## [1] 604.32

pha <- n*(1-p)
pha

## [1] 654.68

=> 1259 * .48 > 10 and 1259 * 0.52 > 10 so it is true.

A news piece on this survey’s findings states, “Majority of Americans think marijuana should be legalized.” Based on your confidence interval, is this news piece’s statement justified?

=> There is evidence towards the opposite. However, it’s possible considering the upper tail of our confidence interval 0.5075972 is above 50%.

Legalize Marijuana, Part II. (6.16, p. 216) As discussed in Exercise above, the 2010 General Social Survey reported a sample where about 48% of US residents thought marijuana should be made legal. If we wanted to limit the margin of error of a 95% confidence interval to 2%, about how many Americans would we need to survey?

a <-(p * (1-p)) / (0.02 / 1.96)^ 2
a

## [1] 2397.158

=> at least 2397 Americans

Sleep deprivation, CA vs. OR, Part I. (6.22, p. 226) 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.

ca <- .08
n_ca <- 11545
or <- .088
n_or <- 4691
se_ca_or <- sqrt(((ca * (1 - ca)) / n_ca) + ((or * (1 - or) / n_or)))
me <- 1.96 * se_ca_or
lbound <- 0.088 - 0.08 - me
ubound <- 0.088 - 0.08 + me
c(lbound, ubound)

## [1] -0.001498128  0.017498128

Barking deer. (6.34, p. 239) 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

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

=> Barking deer doesn’t prefer to certain habitats, barking deer prefer certain habitats over others for forage.

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

=> A chi-square test may be used to answer.

Check if the assumptions and conditions required for this test are satisfied.

=> Independent of each case that contributes a count to the table must be independent of all the other cases in the table. Sample size / distribution. Each scenario must have at least 5 instance.

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

obs_reported <- c(4, 16, 67, 345)
obs_ratio <- c(0.048*426, 0.147*426, 0.396*426, 0.409*426)
chi_deer <- sum((obs_reported - obs_ratio ) ^ 2 / obs_ratio)
p_value_d <- 1 - pchisq(chi_deer, 3)
p_value_d

## [1] 0

Coffee and Depression. (6.50, p. 248) Researchers conducted a study investigating the relationship between caffeinated coffee consumption and risk of depression in women. They collected data on 50,739 women free of depression symptoms at the start of the study in the year 1996, and these women were followed through 2006. The researchers used questionnaires to collect data on caffeinated coffee consumption, asked each individual about physician-diagnosed depression, and also asked about the use of antidepressants. The table below shows the distribution of incidences of depression by amount of caffeinated coffee consumption.

                 Caffeinated coffee consumption
  ≤ 1            2-6             1              2-3            ≥ 4
cup/week      cups/week       cup/day        cups/day        cups/day      Total

Yes 670 373 905 564 95 2,607 No 11,545 6,244 16,329 11,726 2,288 48,132 Total 12,215 6,617 17,234 12,290 2,383 50,739

What type of test is appropriate for evaluating if there is an association between coffee intake and depression?

=> A chi-square test is best suited for the find out to determine the association between coffee intake and deppression.

Write the hypotheses for the test you identified in part (a).

=>There is no association between coffee intake and depression

Calculate the overall proportion of women who do and don’t suffer from depression.

propOfWomenWithDep = 2607/50739
propOfWomenWithoutDep = 1 - propOfWomenWithDep
propOfWomenWithDep

## [1] 0.05138059

propOfWomenWithoutDep

## [1] 0.9486194

Identify the expected count for the highlighted cell, and calculate the contribution of this cell to the test statistic, i.e. (Observed − Expected)2 / Expected).

groups = 5
dF = 4
expected = propOfWomenWithDep*6617
cellFactor = (373 - expected)^2/expected
cellFactor

## [1] 3.205914

The test statistic is χ2 = 20.93. What is the p-value?

pValue = pchisq(20.93,dF,lower.tail = FALSE)
pValue

## [1] 0.0003269507

What is the conclusion of the hypothesis test?

=> The null hypotheses is rejected that there is no association between the coffee intake and depression.

One of the authors of this study was quoted on the NYTimes as saying it was “too early to recommend that women load up on extra coffee” based on just this study. Do you agree with this statement? Explain your reasoning.

=> Agree, there are many ordinal levels of clinical depression which we didn’t take into account, which may shed more light on caffeine or depression.

Rpubs => https://rpubs.com/gunduzhazal/822485

Hazal Gunduz

Chapter 6 - Inference for Categorical Data