Download W2A3 Mean=The sum of all values/n Median=(n+1)/2 Mode=the

W2A3
Mean=The sum of all values/n
Median=(n+1)/2
Mode=the value that appears most often
Range=highest value-lowest value
Deviation of each value=value – mean
z=value-mean/standard deviation
Pk = k*(n+1)/100
Pk = n*k/100
S squared is the Sample variance = [the sum of (the values) squared minus (the sum of the
values) squared/n] /n-1
S is the Standard deviation=the square root of the sample variance or the square root of [the
sum of (the values) squared minus (the sum of the values) squared/n] /n-1
1. For a particular sample of 63 scores on a psychology exam, the following results were
obtained.
First quartile = 57 Third quartile = 72 Standard deviation = 9 Range = 52
Mean = 72 Median = 68 Mode = 70 Midrange = 57
Answer each of the following:
I.
What score was earned by more students than any other score? Why?
II.
What was the highest score earned on the exam?
III.
What was the lowest score earned on the exam?
IV.
According to Chebyshev's Theorem, how many students scored between 54 and 90?
V.
Assume that the distribution is normal. Based on the Empirical Rule, how many students
scored between 54 and 90?
Mean=The sum of all values/n Mean = 72 n=63
72=The sum of all values/63
4536=The sum of all values
Range=highest value-lowest value
52=H-L(yes)
Midrange=lowest value + highest value/2
57=L+H/2
114=L+H (yes)
Q1=57 Q2=68 Q3=72
(31),(39),(48),57, 57,(66), 70,70,70, 72, (75),(83)
I. 70. The mode is 70, therefore it was the score earned by more students.
II. The highest score, take the median and add ½ the range.
Highest score=median+1/2 range=57+52/2=57+26=83
Highest score=83
III. The lowest score, take the median and subtract ½ the range.
Lowest score=57-52/2=57-26=31
Lowest score=31
IV. 100%. Probability of percentage will fall between 54 and 90 = 1-(1/k squared)
First we need to find k
mean – k*standard deviation=54
72-k*9=54
9k=72-54
9k=18
K=2
mean + k*standard deviation= 90
72+k*9=90
9k=90-72
9k=18
K=2
1 - (1/2 squared) = 1 - (1/4) = 1 - .25 = .75
75% of the scores will fall between 54 and 90 That would be 47 students
V. According to the Empierical rule, if k=2 then 95% of all values fall within 2 standard
deviations of the mean. Answer is 95%. That would be 60 students.
2. Find the range, standard deviation, and variance for the following sample data:
87, 75, 58, 29, 21, 19, 80, 27, 61, 77, 26, 16, 79, 1, 68, 100 (Points : 6)
First arrange the data in numerical order.
1, 16, 19, 21, 26, 27, 29, 58, 61, 68, 75, 77, 79, 80, 87,100
Range = highest value – lowest value = 100-1 = 99
Mean = The sum of all values/n =
1+16+19+21+26+27+29+58+61+68+75+77+79+80+87+100=824
824/16 = 51.5
Value (value-mean) (value-mean)squared
1
-50.5
2550.25
16
-35.5
1260.25
19
-32.5
1056.25
21
-30.5
1260.25
26
-25.5
0650.25
27
-24.5
0600.25
29
-22.5
0506.25
58
06.5
0000.25
61
09.5
0090.25
68
16.5
0272.25
75
23.5
0552.25
77
25.5
0650.25
79
27.5
0756.25
80
28.5
0812.25
87
35.5
1260.25
100
48.5
2352.25
Sample Variance= the sum of (deviations squared)/n-1
14630/16-1=14630/15=975.33
Variance = 975.33
Standard deviation = the square root of 975.33 = 31.23
3. In terms of the mean and standard deviation:
-
What does it mean to say that a particular value of x has a standard score of -2.3?
-
What does it mean to say that a particular value of x has a z-score of +1.0?
4. A student scored 81 percent on a test, and was in the 77th percentile. Explain these two
numbers.
The student got 81 percent of the answers correct on the test.
The student did better than 77 percent of the students who took the test.
5. In each of the four examples listed below, one of the given variables is independent (x)
and one of the given variables is dependent (y). Indicate in each case which variable is
independent and which variable is dependent.
I. Ability to concentrate; Level of fatigue
II. Taxi fare; Length of ride
III. Size of package; Cost to ship
IV. DNA of parents; DNA of offspring
6. A sample of purchases at the local convenience store has resulted in the following sample
data, where x = the number of items purchased per customer and f = the number of customers.
x
1
2
3
4
5
f
10
18
8
8
7
·
What does the 18 stand for in the above table?
·
Find the midrange of items purchased.
·
How many items were purchased by the customers in this sample?
The value 18 means that 18 customers purchased two items
Midrange = highest number + lowest number / 2
5+1/2 = 6/2 = 3
Midrange of items purchased = 3
(10*1)+(18*2)+(8*3)+(8*4)+(7*5) = 10+36+24+32+35 = 137
137 items were purchased in this sample
7. Scores on the Stanford-Binet Intelligence scale have a mean of 100 and a standard
deviation of 16, and are presumed to be normally distributed. A person who scores 84 on this
scale has what percentile rank within the population? Show all work as to how this is obtained.
Pk = n*k/100=84*k/100
K = 1.19 round up to 2
mean + k*standard deviation=84
100 + k*16=84
16k=100-84
16k=16
K=1
Let R be the percentile rank. Then,
P(X < 68) = R/100
P(x < 68) = P(Z < (68-100)/16) = 0.0228
R/100 = 0.0228
R = 2.28
Percentile score is 2.
8. SAT I scores around the nation tend to have a mean scale score around 500, a standard
deviation of about 100 points, and are approximately normally distributed. What SAT I score
within the population would have a percentile rank of approximately 2.5? Show all work as to
how this is obtained.
Pk=n*2.5/100
mean+k*standard deviation
500 + 2.5*100 = 500 + 250 = 750