5 questions

QUESTION 1.

The brain weight of adults Swedish males normally distributed with a mean of 1,350g and a standard deviation of 95g. Let Y be the brain weight of a randomly chosen person from this population. Calculate

(i). P(Y≤1,400) (4 points)

(ii). P(1,325≤Y≤1,400) (4 points)

(iii). P(Y≥1,325) (4 points)

(iv). P(Y≥1,450) (4 points)

(v). P(1,450≤Y≤1,500) (4 points)

(v). P(1,300≤Y≤1,350) (4 points)

Note: Draw appropriate normal curves to represent the areas under the curves for (i)-(v).

QUESTION 2.

The serum cholesterol level of a 17 year-old follows a normal distribution with a mean of 176mg/dLi and a variance of 900mg/dLi . What percentages of 17-year olds have serum cholesterol values:

(i). 170 or more? (4 points)

(ii). 166 or less? (4 points)

(iii). 212 or less? (4 points)

(iv). 123 or more? (4 points)

(v). between 176 and 206? (4 points)

(vi).between 123 and 157? (4 points)

(vii).between 155 and 186?(4 points)

Note: Draw appropriate normal curves to represent the areas under the curves for (i)-(vii).

QUESTION 3.

(a).The random variable X is normally distributed with a mean of µ and a standard deviation of σ. It is known that P(X≤66)=0.0359 and P(X≥81)=0.1151.

(i). Give a clearly labelled sketch to represent these probabilities on a normal curve. (2 points)

(ii).Find the values of µ and σ . (10 points)

(iii).Find P(X≤69) (3 points)

(iv). Find P(X>83) (3 points)

(v). Find P(69≤X≤83) (3 points)

Note: Draw appropriate normal curves to represent the areas under the curves for (iii)-(v).

(b). Find the values of the following Zα.notations

(i). Z0.1 (1 point)

(ii). Z0.01 (1 point)

(iii). Z0.025 (1 point)

(iv). Z0.05 (1 point)

(v). Z0.005 (1 point)

QUESTION 4.

(a).(i). Explain why assessing the normality of a random variable often important (2 points)

(II).Under what circumstances is using a normal probability plot to assess the normality of a variable usually better. (2 points)

(iii).Explain in detail what a normal probability plot is and how is it used to assess the normality of a variable. (2 points)

(iv).How is a normal probability plot used to detect outliers? (2 points)

(b).Consider the following three Normal Probability Plots. Plot (i) represents a sample of the final exam scores in a large Introductory Statistics Course. Plot (ii) represents the cell phone rates in February 2003 from different cell phone providers. Plot (iii) represents the amount spent on nonalcoholic beverages for a random sample of 12 consumers from a large number of consumers. Below is the data for the three plots.

Data(i) 88 85 90 81 67 82 63 96 64 39 89 100 76 75 90 70 86 34 84 96

Data(ii) 40 70 60 110 30 70 90 60 35 30 60 80 70 50 75

Data(iii) 361 259 259 176 281 249 184 240 194 265 273 258

In each of the plots identify any possible outliers if present by stating the actual data point(s) and assess the normality of the variable under consideration.

(12 points)

[Hint: Use all the necessary techniques discussed in class to help you answer the question above. I mean use the 70% rule to determine linearity of the normal probability plot, and perform the 1.5IQR calculations to help you identify potential lower and upper outliers in the data sets.]

Plot (i)