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• Normal Distribution

## Normal Distribution

1. What does the standard deviation measure? (2 points)
The standard deviation measures the amount of dispersion in the data. Greater the standard deviation, higher the deviation from the mean.
2. List the characteristics of the binomial distribution. (3 points)
The binomial distribution is a discrete probability distribution which measures the probability of number of success in “n” independent experiments, in which each event has a success probability of “p” each.
3. What is the central limit theorem? Why is the central limit theorem important?
Central limit theorem states that the distribution of sample means of large sample sizes drawn from a population will approximately form a normal distribution. It is necessary as it used in many cases to draw inferences about sample data.
4. List the characteristics of the normal distribution.
The normal distribution has a bell curved shape, which is symmetric, whose mean, median and mode are equal to each other.
Frequency Distribution
Consider the following frequency distribution for a group of 25 people.
Income (thousands of dollars) Number of People
20 – 24 5
25 – 29 6
30 – 34 8
35 – 39 3
40 – 44 3
5. Draw an ogive (also called acumulative frequency polygon). Label your axes. 6. Draw a histogram. Label your axes and discuss skewness. The histogram has a fatter left tail compared to a right tail. Therefore, the distribution is right skewed and has positive skew.
7. To estimate the mean family income in Wisconsin, a researcher collects information from 500 Wisconsin households. State the
a. Individual
Each Wisconsin household
b. Variable
Family income
c. Population
Set of all Wisconsin households
d. Sample
Set of 500 Wisconsin households
e. Parameter
Mean family income in Wisconsin in the set of 500 Wisconsin households in the sample
f. Statistic
Mean family income in Wisconsin in all Wisconsin households
8. A recent poll showed that 42% of Americans do not believe in Darwin’s Theory of Evolution. If 15 Americans are randomly selected, what is the probability that
a. exactly 5Americans do not believe in evolution?
We can use the binomial distribution to compute the probability
P(X = 5, N = 15, p = 0.42) = nCx*px*(1-p)N-x = 0.169
b. at least 9 Americans do not believe in evolution?
P(X >=9, N = 15, p = 0.42) = ΣnCx*px*(1-p)N-x = 0.125
9. What sampling method (simple random, cluster, stratified, convenience, systematic) is being used in each of the following scenarios:
a. Dependents at two bases are studied in depth to determine satisfaction with command support during transitions throughout the military.
Convenience sampling
b. All students are given a number and then a random number generator is used to select five hundred of them to take a survey.
Simple random sampling
c. Every 30th coffee maker is taken off of the assembly line and checked for quality.
Systematic sampling
d. Students are divided by age and then different groups are randomly selected from each group.
Cluster sampling
10. Say which level of measure (interval, ordinal, ratio, nominal) is being used and state the center measures that could be used for each of the following:
a. Marital status (single, married, divorced, widowed) of applicants
Nominal
b. Distance between two cities
Ratio
c. Water temperature measured Fahrenheit
Interval
d. Evaluation of performance by poor, fair, good, excellent
Ordinal
11. A sample of days at a used car dealership reported the following number of cars sold each day:
 5 4 7 3 1 4 3 2 3 9

Calculate the following:
Mode = 3 cars sold per day
Median = (3+4)/2 = 3.5cars sold per day
Mean = 41/10 = 4.1 cars sold per day
Range = 9 – 1 = 8 cars sold per day
Standard deviation = 2.378 cars sold per day
a. How many results are there for the top 3 horses in a race of 10 horses if the winning horses are categorized by first, second, and third?
Number of possibilities = 10*9*8= 10*72 = 720
b. How many ways are there to select a committee of four people from a group of thirty workers?
Number of ways to select = 30C4 = 27,405.
c. How many license plates are there that have two letters followed by five digits?
Number of license plates = 26*26*10*10*10*10*10 = 67,600,000
13. A recent poll showed that 42% of Americans do not believe in Darwin’s Theory of Evolution. If 15 Americans are randomly selected,
a. What is the probability that exactly 6 Americans do not believe in evolution?
We can use the binomial distribution to compute the probability
P(X = 6, N = 15, p = 0.42) = nCx*px*(1-p)N-x = 0.204
b. What is the probability that at least 9 Americans do not believe in evolution?
P(X >=9, N = 15, p = 0.42) = ΣnCx*px*(1-p)N-x = 0.125
14. Say whether the data is qualitative, quantitative discrete, quantitative continuous
a. Weight of a tomatoes sold at the commissary
Quantitative continuous
b. Eye color of students in a statistics class
Qualitative variable
c. Number of hot dogs sold daily at Nathans in New York City
Quantitative discrete
15. A study of families that have televisions determined the number of televisions per households and their probability. The probability that a family had 1,2,3,4 televisions was 0.32, 0.49, 0.14, and 0.05 respectively.
a. Construct a probability distribution.
 X P(x = X) 1 0.32 2 0.49 3 0.14 4 0.05
b. Draw a graph for the probability distribution. c. Find the mean (also called the expected value)
Mean = 1*0.32 + 2*0.49 + 3*0.14 + 4*0.05 = 1.92
d. Find the standard deviation
Standard deviation = SQRT(E(X2) - E(X)2)
E(X2) = 4.34
E(X) = 1.92
Standard deviation = SQRT(4.34 – 1.922) = SQRT(0.6536) = 0.8084
##### Standard Distribution
The amount of juice that a machine puts into a container at a bottling plant is normally distributed with a mean of 11.9 ounces and a standard deviation of 0.3 ounces.
16. What is the probability of selecting a single container that has between 11.9 and 12.6 ounces of juice?
We can compute the Z-values to figure out the probability
Z = (X – Mean)/SD
P (11.9 < X < 12.6) = P ((11.9 – 11.9)/0.3 < Z < (12.6 – 11.9)/0.3) = P(0 < Z < 7/3) = 0.49
17. How much juice is in the fullest 3% of the containers?
We need to compute the value of amount of juice in the top 3% percentile.
Value of Z corresponding to top 3% percentile is 1.88
X = Mean + Z*SD = 11.9 + 1.88*0.3 = 12.464 ounces
18. If a sample of 20 containers is chosen, what is the probability that the mean amount of juice in the containers is more than 12.0 ounces?
Mean amount of juice of 20 containers = 11.9 ounces
Standard deviation of amount of juice of 20 containers = 0.3/sqrt(20) = 0.067
P(Mean amount > 12.0) = P(Z > (12 – 11.9)/0.067) = P(Z > 1.491) = 0.068
19. Consider a jar that contains.
Two yellow tokens numbered 1 and 2
Three green tokens numbered 1,2, and 3
Five red tokens numbered 1,2,3,4 and 5
One token is drawn from the jar:
a. Are the events of drawing a number 2 token or a red token mutually exclusive? Why or why not?
No, the two events are not mutually exclusive events, as the two events can occur at the same time.
b. What is the probability of drawing a number 2 token or a red token?
P( Drawing 2 token or a red token) = 7/10 = 0.7
20. A subway leaves the Madrid airport every 20 minutes for Atocha train station in the city center.
a. State the random variable.
b. Find the height of this uniform distribution.
Height of probability, q = 1/20 = 0.05
c. Find the probability of waiting between 10 and 16 minutes.
P (10 < X < 16) = q*(16 – 10) = 0.05*6 = 0.30
d. Find the probability of waiting less than 5 minutes.
P(X < 5) = 5/20 = 0.25
e. Find the probability of waiting 8 minutes exactly.
P(X = 8) = 0
Since, point probabilities in a uniform distribution is equal to zero.
21. In a math class, each quiz counts 12%, the homework counts 10%, and the final exam counts 30% of the course grade. If a student gets grade of 81, 75, 92, 100, and 70 on the quizzes, 100 for the homework, and 91 on the final exam, what does the student earn for the course?
Course score = 0.12*(81+75+92+100+70)+0.1*10+0.3*91 = 78.46