II. STATEMENT OF THE PROBLEM
As part of a longterm study of individuals 65 years of age or older, sociologists and physicians at the Wentworth Medical Center in upstate New York investigated the relationship between geographic location, health status ( healthy or one or more comorbidities), and depression. Random samples of 20 healthy individuals were selected from three geographic locations: Florida, New York, and North Carolina. Then, each was given a standardized test to measure depression (higher test scores indicate higher levels of depression). Similarly, random samples of 20 individuals with one or more comorbidities (arthritis, hypertension, and/or heart ailment) were taken from the three geographic locations. They were also given the standardized test to measure depression. The researches would also like to know the following: a. Using descriptive statistics to summarize the data from the two studies, what are your preliminary observations about the depression scores? b. Using ANOVA on both data sets, what is the hypotheses being tested in both cases? What are your conclusions? c. Use inferences about individual treatment means where appropriate. What are your conclusions? d. Discuss extensions of this study or other analyses that you feel might be helpful
III. DESCRIPTIVE STATISTICS
1. Data
Table 1.1 shows the data of 60 individuals chosen for the study. Medical 1 reflects test scores from a standardized test to measure depression of healthy 20 individuals from Florida, 20 from New York and 20 from North Carolina. Medical 2 indicates the scores from another set of individuals but this time, with comorbidities.
...questions about any of the solutions given below. 1. Explain the difference between a population and a sample. In which of these is it important to distinguish between the two in order to use the correct formula? mean; median; mode; range; quartiles; variance; standard deviation. Solution: A sample is a subset of a population. A population consists of every member of a particular group of interest. The variance and the standard deviation require that we know whether we have a sample or a population. 2. The following numbers represent the weights in pounds of six 7year old children in Mrs. Jones' 2nd grade class. {25, 60, 51, 47, 49, 45} Find the mean; median; mode; range; quartiles; variance; standard deviation. Solution: mean = 46.166.... median = 48 mode does not exist range = 35 Q1 = 45 Q2 = median = 48 Q3 = 51 variance = 112.1396 standard deviation =10.59 3. If the variance is 846, what is the standard deviation? Solution: standard deviation = square root of variance = sqrt(846) = 29.086 4. If we have the following data
34, 38, 22, 21, 29, 37, 40, 41, 22, 20, 49, 47, 20, 31, 34, 66 Draw a stem and leaf. Discuss the shape of the distribution. Solution: 2 3 4 5 6      219200 48714 0197 6
This distribution is right skewed (positively skewed) because the “tail” extends to the right. 5. What type of relationship is shown by this scatter plot?
45 40 35...
...
Student Exploration: Sight vs. Sound Reactions
Vocabulary: histogram, mean, normaldistribution, range, standard deviation, stimulus
Prior Knowledge Questions (Do these BEFORE using the Gizmo.)
Most professional baseball pitchers can throw a fastball over 145 km/h (90 mph). This gives the batter less than half a second to read the pitch, decide whether to swing, and then try to hit the ball. No wonder hitting a baseball is considered one of the hardest things to do in sports!
1. What are some things in your life you must react to quickly? You need to react quickly when you are in danger, and you need to get away. You also need to react quickly when you are in a car so you don’t get hurt
2. In general, do you think you have quick, slow, or average reactions? I think I have relatively quick reactions, because when I am in a car, I can react to things very fast and when there are things that happy quickly, I can follow them.
Gizmo Warmup
A stimulus is something that can cause you to react. A stimulus can be something you see (visual stimulus), something you hear (auditory stimulus), something you touch (tactile stimulus), or something you smell (olfactory stimulus). In the Sight vs. Sound Reactions Gizmo™, you will compare your reactions to visual and auditory stimuli.
To start, check that the Test is Sight. Click the Start button. When you see a red circle, immediately click your mouse. Take the test...
...under a StandardNormal curve
a) to the right of z is 0.3632;
b) to the left of z is 0.1131;
c) between 0 and z, with z > 0, is 0.4838;
d) between z and z, with z > 0, is 0.9500.
Ans : a) z = + 0.35 ( find 0.5 0.3632 = 0.1368 in the normal table)
b) z = 1.21 ( find 0.5 – 0.1131 = 0.3869 in the normal table)
c ) the area between 0 to z is 0.4838, z = 2.14
d) the area to the right of +z = ( 10.95)/2 = 0.025, therefore z = 1.96
3. Given the Normally distributed variable X with mean 18 and standard deviation 2.5, find
a) P(X < 15);
b) the value of k such that P(X < k) = 0.2236;
c) the value of k such that P(X > k) = 0.1814;
d) P( 17 < X < 21).
Ans : X ~ N ( 18, 2.52)
a) P ( X < 15)
P ( Z < (1518)/2.5) = P ( Z < 1.2) = 0.1151 ( 4 decimal places)
b) P ( X < k) = 0.2236
P ( Z < ( k – 18) / 2.5 ) = 0.2236
From normal table, 0.2236 = 0.76
(k18)/2.5 =  0.76, solve k = 16.1
c) P (X > k) = 0.1814
P ( Z > (k18)/2.5 ) = 0.1814
From normal table, 0.1814 = 0.91
(k18)/ 2.5 = 0.91, solve k = 20.275
d) P ( 17 < X < 21)
P ( (17 18)/2.5 < Z < ( 2118)/2.5)
P ( 0.4 < Z < 1.2) = 0.8849 – 0.3446 = 0.5403 ( 4 decimal places)...
...NORMALDISTRIBUTION
1. Find the
distribution:
a.
b.
c.
d.
e.
f.
following probabilities, the random variable Z has standardnormal
P (0< Z < 1.43)
P (0.11 < Z < 1.98)
P (0.39 < Z < 1.22)
P (Z < 0.92)
P (Z > 1.78)
P (Z < 2.08)
2. Determine the areas under the standardnormal curve between –z and +z:
♦ z = 0.5
♦ z = 2.0
Find the two values of z in standardnormaldistribution so that:
P(z < Z < +z) = 0.84
3. At a university, the average height of 500 students of a course is 1.70 m; the standard
deviation is 0.05 m. Find the probability that the height of a randomly selected student is:
1. Below 1.75 m
2. Between 1.68 m and 1.78 m
3. Above 1.60 m
4. Below 1.65m
5. Above 1.8 m
4. Suppose that IQ index follows the normaldistribution with µ = 100 and the standard
deviation σ = 16. Miss. Chi has the IQ index of 120. Find the percentage of people who
have the IQ index below that of Miss. Chi.
5. The length of steel beams made by the Smokers City Steel Company is normally
distributed with µ = 25.1 feet and σ = 0.25 feet.
a. What is the probability that a steel beam will be less than 24.8 feet long?
b. What is the probability that a steel beam will be more than 25.25 feet
long?
c. What is the probability that a steel beam will be between 24.9 and 25.7
feet long?
d. What is the...
...
NormalDistribution
Student’s Name:
Instructorr:
Date ofSubmission:
The table 1 below shows a relationship between actual daily temperatures and precipitation in the month of January 2011. These data was adopted from a meterological station in the states of Alaska, in the United States. Normal distributed data aresymmetric with a single bell shaped peaks. Th maean of the data it significant in indication the point that the peak is likely to occur. In addition, standard deviation indicates the spread, which is usually referred to asthegirth of thebeell shapedcurve (Balakrishnan & Nevzorov, 2003).
Table 1: Relationship between actual daily temperatures and precipitation of January 2011
Date Actual Temperatures Precipitation 
Jan. 1 30 0 
Jan. 2 25 0 
Jan. 3 31 0 
Jan. 4 33 0 
Jan. 5 29 0 
Jan. 6 36 0.26 
Jan. 7 36 0 
Jan. 8 37 0.01 
Jan. 9 32 0.21 
Jan. 10 28...
...
NormalDistributionNormaldistribution is a statistics, which have been widely applied of all mathematical concepts, among large number of statisticians. Abraham de Moivre, an 18th century statistician and consultant to gamblers, noticed that as the number of events (N) increased, the distribution approached, forming a very smooth curve.
He insisted that a new discovery of a mathematical expression for this curve could lead to an easier way to find solutions to probabilities of, “60 or more heads out of 100 coin flips.” Along with this idea, Abraham de Moivre came up with a model that has a drawn curve through the midpoints on the top of each bar in a histogram of normally distributed data, which is called, “Normal Curve.”
One of the first applications of the normaldistribution was used in astronomical observations, where they found errors of measurement. In the seventeenth century, Galileo concluded the outcomes, with relation to the measurement of distances from the star. He proposed that small errors are more likely to occur than large errors, random errors are symmetric to the final errors, and his observations usually gather around the true values. Galileo’s theory of the errors were discovered to be the...
...1. A formal statement that there is an absence of relationship between variables when tested by a researcher is called: (Points : 1) 
Null hypothesis
Type I error
Type II error
Negative interval

2. Bivariate statistics refers to the statistical analysis of the relationship between two variables. (Points : 1) 
True
False

3. Positive relationships between two variables indicate that, as the score of one increases, the score of the other increases. (Points : 1) 
True
False

4. A result that is probably not attributable to chance is: (Points : 1) 
Type I error
Type II error
Statistical significance
In the semiquartile range

5. A score that is likely to fall into the middle 68% of scores of a normaldistribution will fall inside these values: (Points : 1) 
. +/ 3 standard deviations
+/ 2 standard deviations
+/ 1 standard deviation
semiquartile range

6. It is important to assess the magnitude or strength of a relationship because this assists you with deciding whether or not a variable A causes variable B. (Points : 1) 
True
False

7. In a negative relationship, as the score of one variable decreases, the score on the second variable...
...NormalDistribution: A continuous random variable X is a normaldistribution with the parameters mean and variance then the probability function can be written as
f(x) =  < x < ,  < μ < , σ > 0.
When σ2 = 1, μ = 0 is called as standardnormal.
Normaldistribution problems and solutions – Formulas:
X < μ = 0.5 – Z
X > μ = 0.5 + Z
X = μ = 0.5
where,
μ = mean
σ = standard deviation
X = normal random variable
NormalDistribution Problems and Solutions – Example Problems:
Example 1:
If X is a normal random variable with mean and standard deviation calculate the probability of P(X<50). When mean μ = 41 and standard deviation = 6.5
Solution:
Given
Mean μ = 41
Standard deviation σ = 6.5
Using the formula
Z =
Given value for X = 50
Z =
=
= 1.38
Z = 1.38
Using the Z table, we determine the Z value = 1.38
Z = 1.38 = 0.4162
If X is greater than μ then we use this formula
X > μ = 0.5 + Z
50 > 41 = 0.5 + 0.4162
P(X) = 0.5 + 0.4162
= 0.9162
Example 2:
If X is a normal random variable with mean and standard deviation calculate the probability of P(X< 37). When mean μ = 20 and standard deviation = 15
Solution:
Given
Mean μ = 20...
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