NOTE: For all problems in this assignment, do NOT use the empirical rule unless specifically told to do so. 

A rural area on a nearby interstate is famous for drivers traveling at excessive speeds. A state trooper decides to investigate. Using radar, they measure the speed of a few hundred vehicles. After graphing and doing some simple calculations, they report that the speeds (in miles per hour) were approximately normally distributed, N(72, 6.4). 

For your calculations, use two decimal places. You can round the second decimal place. For example, 74.8862 can be rounded to 74.89.

  1. In your own words, describe what is meant by standard deviation. Note: I do not want a mathematical explanation. Rather, I want you to explain in words what the standard deviation is. You can include numbers if you need them to help with your discussion, but I do not want to see the actual formula for SD. This question is worth up to 3 points. You will be given full points based on your ability to convince the grader that you do indeed understand the concept of SD. 
  2. One speed was recorded as had a score of 85 miles per hour. What percentile does this correspond to? Hint: Remember that R’s pnorm() function gives the area under the density curve. 
  3. What speed would be recorded to be at the 95th percentile? 
  4. What percentage of drivers were traveling at a “respectable” speed of between 65 and 75 miles per hour?
  5. What percentage of drivers were traveling at a speed with a z-score of lower than -1.5? 
  6. What percentage of drivers were traveling at a speed with a z-score of greater than +2? 

Problem #2

The time for a machine to manufacture a part for a regulator for scuba diving (in minutes) is estimated using the data set shown here. For your convenience, the times have been sorted. 

12.7, 12.8, 13, 13.2, 13.3, 13.3, 13.3, 13.4, 13.4, 13.4, 13.5, 13.5, 13.5, 13.5, 13.5, 13.5, 13.5, 13.5, 13.5, 13.6, 13.6, 13.6, 13.6, 13.6, 13.6, 13.6, 13.7, 13.7, 13.7, 13.7, 13.7, 13.7, 13.8, 13.8, 13.8, 13.8, 13.8, 13.8, 13.8, 13.9, 13.9, 13.9, 13.9, 13.9, 13.9, 13.9, 13.9, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14.1, 14.1, 14.1, 14.2, 14.2, 14.2, 14.2, 14.2, 14.2, 14.2, 14.2, 14.3, 14.3, 14.3, 14.3, 14.3, 14.3, 14.3, 14.3, 14.3, 14.3, 14.4, 14.4, 14.4, 14.4, 14.5, 14.5, 14.5, 14.5, 14.5, 14.5, 14.6, 14.6, 14.6, 14.7, 14.7, 14.7, 14.7, 14.8, 14.8, 14.8, 14.8, 14.9, 14.9, 14.9, 14.9, 15, 15, 15, 15.1, 15.1, 15.2, 15.3, 15.3, 15.4, 15.5, 15.5, 15.5, 15.7, 15.8, 16, 17.8

Using R (as well as your mind!), answer the following:

  1. Show/Describe the distribution of this variable. (Hint: Review in your notes/resources for this course what is some key information needed to properly describe a distribution.)
  2. Draw a boxplot. What does an open circle represent? What do you want to do with any such observation(s)? 
  3. Take a look at the histogram. Do you think this variable normally distributed? Let’s assume that we think it is. There is another graph we can use to help confirm that the variable is normally distributed. Plot this graph and describe what this graph tells us. 

Problem #3

You must use the empirical rule to answer the problems in this question. In other words, I do not want you to use R’s pnorm() function to answer any of these questions. 

Data from the manufacturing of a different piece of scuba equipment was approximately normally distributed in minutes: N(6.2, 0.7)

  1. One item took 4.8 minutes. What percentile does this correspond to? 
  2. What percentage of items took between 4.8 minutes and 7.6 minutes to manufacture?
  3. What percentage of items were made in less than 4.1 minutes? 
  4. One item was manufactured with a z-score of +2. How many minutes did it take to make this item? 
  5. Do not use the empirical rule for this problem. Instead, use R for a precise calculation. One item was in the 72nd percentile. What time does that correspond to? Hint: There is an R function that will give you this exact piece of information. That is, it converts the area under the normal density curve back into a z-score. 

Problem #4

Here is a dataset in which the height of a group of school children was compared to their weights to see if there is a relationship. 

Here are the observations: 

Height (feet) Weight (pounds)
2.79 11.6
3.9 21.9
5.5 27.6
1.65 3.6
4.43 23
4.51 23.5
3.43 16
3.45 16.2
3.8 16
3.11 13.8
3.52 16.7
3 13

Create a scatterplot in which the explanatory variable is the height and the response variable is the weight. You must have all of your R code saved in a script called: height_weight.R

For your convenience, I am providing you with the vectors to create one of the variables. You can paste it into your R script:

height <- c(2.79, 3.9, 5.5, 1.65, 4.43, 4.51, 3.43, 3.45, 3.8, 3.11, 3.52, 3)

Generate the other vector using the table above. 

When you are done, your graph should look something like this:

Chart, scatter chart

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