Chapter 2: Summarizing dataContent check — Statistics (OpenIntro), Chapter 2: Summarizing data

Answers may vary. Scatterplots are helpful in quickly spotting associations relating variables, whether those associations come in the form of simple trends or whether those relationships are more complex.

Consider the case where your vertical axis represents something “good” and your horizontal axis represents something that is only good in moderation. Health and water consumption fit this description: we require some water to survive, but consume too much and it becomes toxic and can kill a person.

$x_1$ corresponds to the interest rate for the first loan in the sample (10.90%), $x_2$ to the second loan’s interest rate (9.92%), and $x_i$ corresponds to the interest rate for the $i^{th}$ loan in the data set. For example, if $i=4$⁠, then we’re examining $x_4$⁠, which refers to the fourth observation in the data set.

The sample size was $n=50$⁠.

The skew is visible in all three plots, though the flat dot plot is the least useful. The stacked dot plot and histogram are helpful visualizations for identifying skew.

The interest rates for individual loans.

There might be two height groups visible in the data set: one of the students and one of the adults. That is, the data are probably bimodal.

Figure 2.9 shows three distributions that look quite different, but all have the same mean, variance, and standard deviation. Using modality, we can distinguish between the first plot (bimodal) and the last two (unimodal). Using skewness, we can distinguish between the last plot (right skewed) and the first two. While a picture, like a histogram, tells a more complete story, we can use modality and shape (symmetry/skew) to characterize basic information about a distribution.

Since $Q_1$ and $Q_3$ capture the middle 50% of the data and the median splits the data in the middle, 25% of the data fall between $Q_1$ and the median, and another 25% falls between the median and $Q_3$⁠.

These visual estimates will vary a little from one person to the next: $Q_1=$ 8%, $Q_3=$ 14%, $\text{IQR}=Q_3-Q_1=6\%$⁠. (The true values: $Q_1=7.96\%$⁠, $Q_3=13.72\%$⁠, $\text{IQR}=5.76\%$⁠.)

Mean is affected more.

Standard deviation is affected more. Complete explanations are provided in the material following Guided Practice 2.17.

Answers will vary! If we’re looking to simply understand what a typical individual loan looks like, the median is probably more useful. However, if the goal is to understand something that scales well, such as the total amount of money we might need to have on hand if we were to offer 1,000 loans, then the mean would be more useful.

Note: answers will vary. There is some correspondence between high earning and metropolitan areas, where we can see darker spots (higher median household income), though there are several exceptions. You might look for large cities you are familiar with and try to spot them on the map as dark spots.

Positive association: mammals with longer gestation periods tend to live longer as well.

Association would still be positive.

No, they are not independent. See part (a).

The graph below shows a ramp up period. There may also be a period of exponential growth at the start before the size of the petri dish becomes a factor in slowing growth.

Population mean, ${\mu }_{2007}=52$⁠; sample mean, ${\bar{x}}_{2008}=58$⁠.

Population mean, ${\mu }_{2001}=3.37$⁠; sample mean, ${\bar{x}}_{2012}=3.59$⁠.

Any 10 employees whose average number of days off is between the minimum and the mean number of days off for the entire workforce at this plant.

Dist 2 has a higher mean since $20>13$⁠, and a higher standard deviation since 20 is further from the rest of the data than 13.

Dist 1 has a higher mean since $-20>-40$⁠, and Dist 2 has a higher standard deviation since -40 is farther away from the rest of the data than -20.

Dist 2 has a higher mean since all values in this distribution are higher than those in Dist 1, but both distribution have the same standard deviation since they are equally variable around their respective means.

Both distributions have the same mean since they’re both centered at 300, but Dist 2 has a higher standard deviation since the observations are farther from the mean than in Dist 1.

About 30.

Since the distribution is right skewed the mean is higher than the median.

Q1: between 15 and 20, Q3: between 35 and 40, IQR: about 20.

Values that are considered to be unusually low or high lie more than 1.5⁠$\times$⁠IQR away from the quartiles. Upper fence: Q3 + 1.5 $\times$ IQR = $37.5+1.5\times 20=67.5$⁠; Lower fence: Q1 - 1.5 $\times$ IQR = $17.5-1.5\times 20=-12.5$⁠; The lowest AQI recorded is not lower than 5 and the highest AQI recorded is not higher than 65, which are both within the fences. Therefore none of the days in this sample would be considered to have an unusually low or high AQI.

The histogram shows that the distribution is bimodal, which is not apparent in the box plot. The box plot makes it easy to identify more precise values of observations outside of the whiskers.

The distribution of number of pets per household is likely right skewed as there is a natural boundary at 0 and only a few people have many pets. Therefore the center would be best described by the median, and variability would be best described by the IQR.

The distribution of number of distance to work is likely right skewed as there is a natural boundary at 0 and only a few people live a very long distance from work. Therefore the center would be best described by the median, and variability would be best described by the IQR.

The distribution of heights of males is likely symmetric. Therefore the center would be best described by the mean, and variability would be best described by the standard deviation.

The median is a much better measure of the typical amount earned by these 42 people. The mean is much higher than the income of 40 of the 42 people. This is because the mean is an arithmetic average and gets affected by the two extreme observations. The median does not get effected as much since it is robust to outliers.

The IQR is a much better measure of variability in the amounts earned by nearly all of the 42 people. The standard deviation gets affected greatly by the two high salaries, but the IQR is robust to these extreme observations.

The distribution is unimodal and symmetric with a mean of about 25 minutes and a standard deviation of about 5 minutes. There does not appear to be any counties with unusually high or low mean travel times. Since the distribution is already unimodal and symmetric, a log transformation is not necessary.

Answers will vary. There are pockets of longer travel time around DC, Southeastern NY, Chicago, Minneapolis, Los Angeles, and many other big cities. There is also a large section of shorter average commute times that overlap with farmland in the Midwest. Many farmers’ homes are adjacent to their farmland, so their commute would be brief, which may explain why the average commute time for these counties is relatively low.

0.451 represents the proportion of individual applicants who have a mortgage.

0.802 represents the fraction of applicants with mortgages who applied as individuals.

0.122 represents the fraction of joint borrowers who own their home.

0.135 represents the home-owning borrowers who had a joint application for the loan.

Answers may vary a little. The counties with population gains tend to have higher income (median of about $45,000) versus counties without a gain (median of about $40,000). The variability is also slightly larger for the population gain group. This is evident in the IQR, which is about 50% bigger in the gain group. Both distributions show slight to moderate right skew and are unimodal. The box plots indicate there are many observations far above the median in each group, though we should anticipate that many observations will fall beyond the whiskers when examining any data set that contain more than a couple hundred data points.

Answers will vary. The side-by-side box plots are especially useful for comparing centers and spreads, while the hollow histograms are more useful for seeing distribution shape, skew, and potential anomalies.

We see the order of the categories and the relative frequencies in the bar plot.

There are no features that are apparent in the pie chart but not in the bar plot.

We usually prefer to use a bar plot as we can also see the relative frequencies of the categories in this graph.

The vertical locations at which the ideological groups break into the Yes, No, and Not Sure categories differ, which indicates that likelihood of supporting the DREAM act varies by political ideology. This suggests that the two variables may be dependent.

We would be assuming that these two variables are independent.

The study is an experiment, as patients were randomly assigned an experiment group. Since this is an experiment, the results can be used to evaluate a causal relationship between the malaria vaccine and whether patients showed signs of an infection.

$4/6-7/14=0.167$ or about 16.7% in favor of the vaccine. This difference due to chance is much smaller than the difference observed in the actual groups.

(i) False. Instead of comparing counts, we should compare percentages of people in each group who suffered cardiovascular problems. (ii) True. (iii) False. Association does not imply causation. We cannot infer a causal relationship based on an observational study. The difference from part (ii) is subtle. (iv) True.

Proportion of all patients who had cardiovascular problems: $\frac{\mathrm{7,979}}{\mathrm{227,571}}\approx 0.035$

The expected number of heart attacks in the rosiglitazone group, if having cardiovascular problems and treatment were independent, can be calculated as the number of patients in that group multiplied by the overall cardiovascular problem rate in the study: $\mathrm{67,593}\ast \frac{\mathrm{7,979}}{\mathrm{227,571}}\approx$ $2370$⁠.

(i) $H_0$⁠: The treatment and cardiovascular problems are independent. They have no relationship, and the difference in incidence rates between the rosiglitazone and pioglitazone groups is due to chance. $H_A$⁠: The treatment and cardiovascular problems are not independent. The difference in the incidence rates between the rosiglitazone and pioglitazone groups is not due to chance and rosiglitazone is associated with an increased risk of serious cardiovascular problems. (ii) A higher number of patients with cardiovascular problems than expected under the assumption of independence would provide support for the alternative hypothesis as this would suggest that rosiglitazone increases the risk of such problems. (iii) In the actual study, we observed 2,593 cardiovascular events in the rosiglitazone group. In the 1,000 simulations under the independence model, we observed somewhat less than 2,593 in every single simulation, which suggests that the actual results did not come from the independence model. That is, the variables do not appear to be independent, and we reject the independence model in favor of the alternative. The study’s results provide convincing evidence that rosiglitazone is associated with an increased risk of cardiovascular problems.

Decrease: the new score is smaller than the mean of the 24 previous scores.

Calculate a weighted mean. Use a weight of 24 for the old mean and 1 for the new mean: $(24\times 74+1\times 64)/(24+1)=$ $73.6$⁠.

The new score is more than 1 standard deviation away from the previous mean, so increase.

No, we would expect this distribution to be right skewed. There are two reasons for this: (1) there is a natural boundary at 0 (it is not possible to watch less than 0 hours of TV), (2) the standard deviation of the distribution is very large compared to the mean.

The distribution of ages of best actress winners are right skewed with a median around 30 years. The distribution of ages of best actor winners is also right skewed, though less so, with a median around 40 years. The difference between the peaks of these distributions suggest that best actress winners are typically younger than best actor winners. The ages of best actress winners are more variable than the ages of best actor winners. There are potential outliers on the higher end of both of the distributions.