Please see the attached file for proper questions with graphs.

Consider the following histogram of the ages (in whole years) of patients with heart disease, collected at the University Hospital, Basel, Switzerland around 1989. Use the histogram to answer the following questions.

Age (years)

Figure 1: Data from Detrano et al. *International application of a new probability algorithm for the diagnosis of coronary artery disease.* 1989.

The first bin of this histogram includes left endpoints and right endpoints and the remaining bins exclude left endpoints and include right endpoints.

a) How many patients are considered in this study, according to the histogram?

(2 points) b) What is the maximum possible value of the mean age of the patients, according to the histogram?

(2 points) c) What is the minimum possible value of the mean age of the patients, according to the histogram? Consider the possible left-exclusions.

(3 points)

***Use R studio for** **Problem 2**

Suppose, in *R*, we have a numeric vector *x* of length five describing the following coordinate values: The first coordinate is 0.809, the second coordinate is 0.983, the third coordinate is 0.630, the fourth coordinate is 0.971 and the last coordinate is 0.832. We may create such a vector using the following *R* code:

x = c (0.809 , 0.983 , 0.630 , 0.971 , 0.832)

a) What are the precise values of the following five expressions in *R*:

mean(x) median(x) min(x) max(x)

range (x)

(4 points) b) Of the above expressions, which are scalars, and which are vectors of length at least two?

(4 points)

Consider the natality statistics in Table 1 for the population of the United States in 1992 reported by the National Center for Health Statistics (a division of the Centers for Disease Control and Prevention in the United States). According to these data, the probability that a person selected uniformly at random among all people who gave birth in the United States in 1992 was in each of the following age groups are as follows:

a) What was the probability that a person who gave birth in the United States in 1992 was 24 years of age or younger?

(1 point)

Age Probability

ă15 0.003 15–19 0.124

20–24 0.263

25–29 0.290

30–34 0.220

35–39 0.085

40–44 0.014 ě45 0.001 Total 1.000

**Table 1**: Natality statistics from USA, 1992.

b) Given that a person who gave birth in the United States in 1992 was under 30 years of age, what was the probability that they were not yet 20 years of age?

(1 point) c) Given that a person who gave birth in the United States in 1992 was 35 years of age or older, what was the probability that they were under 40 years of age? (1 point)

Consider the following numbers from problem 2:

*x* “ p0*.*809*,*0*.*983*,*0*.*630*,*0*.*971*,*0*.*832q*.*

The sample mean of *x* is 0.845. Provide a 0.95 confidence interval for the mean of *x*, assuming the mean is normally distributed.

(2 points)

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