Answer the questions in the document.
Question 2. Let the body temperature of a healthy adult be a normally distributed random variable F which has a mean of 98.6◦F and a standard deviation of 0.62◦F. To convert degrees Fahrenheit to Celsius, we use the formula
C =5/9(F −32)
a) If C is the body temperature of a healthy adult that is measured in degrees Celsius, what is the distribution of C?
b) What is the probability that a healthy adult will have a temperature that exceeds 38◦C?
Question 3. A candy manufacturer produces chocolate bars whose mean weight is 6 ounces with a standard deviation of 0.15 ounces. Assume the weight of the candy bars produced are normally distributed and their weights are independent of one another. Suppose you buy four candy bars. Let the total weight of the chocolate bars be given by, TW = W1 +W2 +W3 +W4, where Wi is the weight of the ith candy bar (i = 1,2,3,4).
a) Determine the distribution of the total weight, TW.
b) What is the probability that the total weight of the four candy bars is less than 23 ounces?
Question 4. Color blindness aﬀects 1 out of every 76 people in the United States. Suppose that you randomly select a resident of the US. Let Y be the random variable that represents that the person selected is color blind.
a) Specify the distribution of Y.
b) What are the mean and the standard deviation of Y?
Question 5. The US National Collegiate Athletic Association (NCAA) sponsors a basketball tournament each spring amongst the most successful, men’s basketball programs. Advertised as “March Madness”, the best 64 teams in Division I are divided into tournament brackets where each team is assigned to play in one of four regions. The strongest team in each region is “seeded”, or ranked, number 1. The weakest team is seeded 16. The ﬁrst round of the tournament always pits the number 1 seed against the number 16 seed. The number 16 seed has (almost) never won a tournament game against the number 1 seed. The dataset for this lab contains the scores of these mismatched games from 1985 to 2015 for each of the four regions. It can be found on the class’ Blackboard account as MarchMadness1985-2015.csv. The variables are deﬁned as follows:
• Year = Year of the tournament
• Region = Regional Bracket
• Seed1 = University team that was seeded ﬁrst in the bracket
• Seed16 = University team that was seeded sixteenth in the bracket
• Score1 = points the ﬁrst seeded team scored in the game against the 16th seed
• Score16 = points the 16th seeded team scores in the game against the 1st seed
• ScoreDiﬀ = number of points the ﬁrst seeded team beat the 16th seeded team by.
a) Construct a histogram of the score diﬀerences. Describe the shape of the histogram. Is it plausible that the diﬀerences could come from a normal distribution? Why or why not?
b) Calculate the mean and the standard deviation for the score diﬀerences using StatCrunch. Round your results to two decimal places.
c) Edit the histogram you created to add the normal density curve. In the histogram command box, under Display options in the Overlay distrib box select Normal, then enter the values of the mean and standard deviation you calculated above. Click Compute. Do you now ﬁnd it plausible that the diﬀerence could come from a normal distribution? Why or why not? Include the histogram with the density curve overlaid in your lab report.
d) Use StatCrunch to construct a normal plot (qqplot) of the score diﬀerences. Select Graph → QQPlot. Select the column for the score diﬀerences. Click Compute. From the QQPlot can we say that the score diﬀerences seem to follow a normal distribution?
e) For the number 16 team to defeat the number 1 seed, what would the score diﬀerence need to be? Using that information, calculate the probability of a number 16 seed beating the number one seed. Report your answer to 4 decimal places.