Social & Cultural Basis for Community and Primary Health Programs

Statistical Reasoning 1, 2019 Quiz 1 Directions: There are a total of 12 questions on this quiz. Each question has only one correct answer choice.

1. Which of the following is true about an observational prospective cohort study being designed to measure the association between cigarette smoking (smoking ≥ 1 pack of cigarettes per day, smoking < 1 pack of cigarettes per day, or no smoking) to be assessed on 3/1/20, and catching a cold between 3/2/20 and 05/31/20?

a. This design minimizes the potential for systematic differences between the exposure groups.

b. Both the exposure and outcome are measured at the same point in time. c. Subjects self-select to be in the exposure groups, so potential confounders will need

to be considered when assessing the outcome/exposure relationship. d. Because this is a prospective study, the risk of confounding is minimized.

2. Which of the following statements defines the 97.5th percentile value for a data sample,

regardless of the shape of the data distribution?

a. The value such that 2.5% of the observations in the sample are larger than this value, and 97.5% of the observations in the sample are less than or equal to this value.

b. The value such that 97.5% of the observations in the sample are larger than this value, and 2.5% of the observations in the sample are less than or equal to this value.

c. The sample mean minus 2 standard deviations. d. The sample mean plus 2 standard deviations.

3. Which of the following sample statistics will tend to decrease (i.e. systematically decrease) with increasing sample size (n)?

a. The mean b. The median c. The standard deviation d. None of the above

You will need the following information to answer questions 4 through 6: This following histogram shows the distribution of (first, for the year) health inspection scores (on a scale of 0-100, with 100 being the highest possible score) on restaurants inspected in Austin, Texas1 (USA) from January through mid-April 2019. All restaurants in Austin are inspected at least once every calendar year. All values shown are from the first inspection made in this time period (a small percentage of restaurants are inspected more than once in a given year).

1 https://data.austintexas.gov/dataset/Restaurant-Inspection-Scores/ecmv-9xxi, last accessed 04/08/19 3:16 pm.

Sample Summary Statistics (n=1,699)) Mean ( ): 92 Standard Deviation (s): 7.0 Range: 56 to 100 Percentiles: 3rd : 72 4th 74 5th : 75 25th : 89 50th : 93 75th :97 95th :100 96th 100 97th :100

4. Which of the following statements best describes the “story” of restaurant inspection

scores in January through mid-April 2019, as told by the histogram, and the summary statistics?

a. The majority of restaurants received relatively high inspection scores (≥ 90)

and a smaller percentage had lower scores than this majority. b. The majority of restaurants received relatively low inspection scores (£ 90)

and a smaller percentage had higher scores than this majority. c. The percentage of restaurants receiving a given score was relatively similar

for all values across the range of inspection scores. d. Since so many restaurants received a perfect score of 100 (at least 5% of those

inspected), it is clear that some restaurant owners were bribing the inspector assigned to them.

5. Assume that the restaurants chosen for inspection in the period January – mid-April 2019

were randomly selected from all restaurants in Austin. Suppose this histogram is remade after the results for inspections performed in late-April through December 2019 are also available. Most likely, what shape will the data in this updated histogram have?

a. uniform b. left skewed (negatively skewed) c. right skewed (positively skewed) d. normal

6. Suppose that restaurants receiving an inspection score of 75 or less are required to have a

follow-up inspection within a month of receiving this score. Approximately, what percentage of inspections required such a follow-up inspection?

a. 10% b. 3% c. 5% d. this cannot be answered without a standard normal table

You will need the following information to answer questions 7-8. The following boxplots show the distribution of restaurant inspection scores (first inspection) for the years 2016-2018, and January- mid-April 2019.

7. In which year was the lowest inspection score reported?

a. 2016 b. 2017 c. 2018 d. 2019

8. You have been recently hired as the health commissioner for the city of Austin. You have received emails from 3 restaurant owners claiming the inspections have been made with inconsistent criteria for the years 2016 -2019. Is there evidence in these data of systematic inconsistency across the years?

a. Yes, because the median score in 2016 was substantially lower than the median score in the other three years.

b. Yes, because the restaurant scores are not (approximately) normally distributed in any of the years.

c. No, because the distributions over the four years are generally similar in terms of shape, median values and the interquartile ranges (25th – 75th percentiles)

d. No, because in each of the four years, some restaurants received perfect inspection scores of 100.

You will need to use the following information to answer questions 9 through 12. A 2018 JAMA2 article details the results from a randomized trial designed and conducted, as per the authors:

“To test the effect of a multicomponent behavioral intervention on child body mass index (BMI, calculated as weight in kilograms divided by height in meters squared) growth trajectories over 36 months among preschool-age children at risk for obesity.”

The follow sections from the article abstract give details on the study design and the intervention being tested:

“DESIGN, SETTING, AND PARTICIPANTS A randomized clinical trial assigned 610 parent-child pairs from underserved communities in Nashville, Tennessee, to a 36-month intervention targeting health behaviors or a school-readiness control. Eligible children were between ages 3 and 5 years and at risk for obesity but not yet obese. Enrollment occurred from August 2012 to May 2014; 36-month follow-up occurred from October 2015 to June 2017. INTERVENTIONS The intervention (n = 304 pairs) was a 36-month family-based, community-centered program, consisting of 12 weekly skills-building sessions, followed by monthly coaching telephone calls for 9 months, and a 24-month sustainability phase providing cues to action. The control (n = 306 pairs) consisted of 6 school-readiness sessions delivered over the 36-month study, conducted by the Nashville Public Library.”

The following graphic shows separate boxplots of the BMI distribution by randomization group at specific times during the study follow-up period:

2

9. Which of the following statements best summarizes the body mass index (BMI) distributions for children randomized to the Intervention groups versus children randomized to the Control treatment group, across the follow-up period? a. At the start of the study (time=0), most of the children (> 75%) randomized to the

control group had BMIs larger in value than the median BMI for children randomized to the control groups.

b. The two BMI distributions are heavily left-skewed at each of the featured time- points.

c. While the medians for the two groups are both similar at each of the times points, and generally consistent over time, the variation and number of outliers in the BMI distributions tend to increase over time.

d. The difference in median BMI between the intervention and control groups consistently increases over the 36-month follow-up period.

10. Approximately, what is the range (minimum to maximum value) of individual BMI values for children in the Control group at the end of the study (t=36 months)? a. 16.0 to 19.0 b. 15.0 to 20.0 c. 14.5 to 23.0 d. 14.5 to 25.5

At the end of the study follow-up period (36 months), the mean and standard deviations of the BMI values (in kg/m2) are as follows:

Intervention: 𝑥"#$%&'%#$()# = 17.8; 𝑠(#$%&'%#$()# = 2.2 Control: 𝑥3)#$&)4 = 17.7; 𝑠3)#$&)4 = 2.0

11. Recall that 306 children were randomized to the control arm of the study. Suppose the researchers had instead had the budget to randomize 600 children (taken from the same population as the original 306) to this control group. How would you expected the standard deviation of these 600 BMI measurements at 36 months to compare to the estimated standard deviation among the original 306 children (2.0)? a. The sample standard deviation of the 600 BMI values will likely be larger in value than

2.0. b. The sample standard deviation of the 600 BMI values will likely be smaller in value

than 2.0. c. The sample standard deviation of the 600 BMI values will likely be similar in value to

2.0 , but there is no way to predict exactly how the two standard deviation estimates will compare in value.

d. The sample standard deviation of the 600 BMI will be approximately half of 2.0 (i.e. 1.0) in value, since the larger sample has about twice as many children as the original control group.

12. The parents of a child randomized to the Intervention group want to know the child’s BMI percentile at 36 months of follow-up (relative to other in the intervention group at 36 months), so that they can compare this to the child’s previous percentile values. At this follow-up time, the child has a BMI of 15.6. Based on the information given above, and assuming that the distribution of BMI measurements in the Intervention group at 36 months of follow-up is (approximately) normal, to what percentile in this distribution does a BMI of 15.6 correspond? a. 16th b. 32nd c. 50th d. 68th e. 84th