Download Statistics - Onteora Central School District

Statistics
Ways to Collect Data ­ MORE DETAILED IN YOUR CLASS NOTES
Survey A benefit ­ you can collect a lot of information easily for a relatively low cost
A weakness ­ motivation for responding (those who are responding might be doing so because of a strong opinion, so the sample may not be a fair representation of the larger population)and/or lack of response
Controlled Experiment ­ remember, control group and experimental group
A treatment is imposed/given.
Concerns: Placebo effect
Observational Study
You observe, document and report. You do not give any type of treatment. Often used when ethics are a concern ­ you can observe what happens after someone is exposed to a harmful chemical, but you cannot give them a harmful chemical to see what happens.
Concerns: these are usually long term, so you can 'lose' subjects and over time, more and more outside influences can play a role in what you are observing.
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Ways to Interpret Data ­ MORE DETAILED IN YOUR CLASS NOTES
Measures of Central Tendency
Mean
Median Mode
Measures of Dispersion
Range
IQR
Variance and Standard Deviation
Five Number Summary
Minimum Value
Q1 (first quartile)
Median (same as Q2)
Q3 (third quartile)
Maximum Value
Remember, Population ­ you must know all data values.
Sample ­ you know a portion of the entire group of data values. 2
Normal Distribution ­
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Percentiles and the Normal Curve
All of this information is on the percentile handout I gave you.
1. Consider an IQ test, with a mean of 100 and a standard deviation of 15. If a
student scores a 115, would percentile would that score represent?
2 An introductory psychology teacher who has taught for years has developed a comprehensive final exam that is normally distributed with a mean of 200 points and a standard deviation of 25 points. (a) What percentage of the students score above 200 points? (b) What percentage of the students score below 175 points? (c) What percentage of the students score more than 250 points? (d) What are the percentile ranks for the three scores 200, 175, and 250?
3. Suppose Emily is taken to the doctor for a well-baby check and it is
determined that she is in the 5th percentile for height and 7th percentile for
weight. What do you now know about Emily, as compared to other children
her age?
4. Given a standardized test with a mean score of 78 and a standard deviation
of 5.9, a student who scored in the 60th percentile could have a score of:
a. 84
b. 72
c. 79
d. 76
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1. The mean is 82.75 and the standard deviation is 2.25. If
the scores were normally distributed, which of the following
scores would be most likely to occur?
d. 77
c. 80.5
b. 87.25
a. 90
2. Given a normal distribution with a mean of 240 and a
standard deviation of 32, what percent of scores would lie
within the following intervals:
a. 240 and 272
b. 192 and 256
c. 176 and 336
d. 160 and 320
3. The company you work for manufactures light bulbs.
They advertise that the “average light bulb” can burn for
1,000 hours. Tests have shown that this is the mean length
of time. The times that the lights can burn are normally
distributed with a standard deviation of 200 hours. What
percent of the bulbs could be expected to last 600 or fewer
hours?
d. 30.9
c. 6.7
b. 2.3
a. 0.6
4. On a standardized test with a normal distribution, the mean is 85
and the standard deviation is 5. If 1,200 students take the exam,
approximately how many of them are expected to earn scores between
90 and 95?
a. 14
b. 98
c. 163
d. 1,172
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5. In a normal distribution, x is the mean and σ is the standard
deviation. If x + 0.5σ = 100 and x ­ 0.5σ = 80, what is the mean?
a. 5
b. 10
c. 20
d. 90
6. A particular tire company advertises a tire that lasts for
80,000 miles. The mileage for the tires is a normal
distribution with a mean of 80,000 miles, and a standard
deviation of 10,000 miles. If the company produces 32,000
tires, how many of them would be expected to last between
65,000 and 100,000 miles?
a. 27,212
b. 29, 120
c. 30,528 d. 31,264
7. On a standardized test with a normal distribution, the
mean was 42 and the standard deviation was 2.6. Which
score could be expected to occur less than 5 percent of the
time?
a. 50
b. 45
c. 39
d. 37
8. You manage a company that manufactures nuts and
bolts. The size of the diameters of the bolts manufactured
produces a normal distribution. The mean size of a certain
bolt is 3 centimeters; with a standard deviation is 0.1
centimeter. Bolts that vary from the mean by more than 0.3
centimeters cannot be sold. If the company manufactures
150,000 of the 3 cm bolts, approximately how many of them
cannot be sold?
a. 150
b. 300
c. 15,000 d. 30,000
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9. In a normal distribution, x is the mean and σ is the
standard deviation.
If x + 2σ = 60, and x ­ 2σ = 40,
what is the standard deviation?
d. 50
c. 20
b. 10
a. 5
10. The mean age of the entering freshman class at a certain
university is 18.5, with a standard deviation of 0.75 years. If
the data produces a normal distribution, find.
a. the percent of students who are between 19.25 and 17.75
years of age
b. the number of students who could be expected to be
younger than x ­ σ years of age, if the total number of
incoming freshman is 1,200 students.
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A set of normally distributed student test scores has a mean of 80
and a standard deviation of 4. Determine the probability that a
randomly selected score will be between 74 and 82.
The amount of time that a teenager plays video games in any given week
is normally distributed. If a teenager plays video games an average of 15
hours per week, with a standard deviation of 3 hours, what is the
probability of a teenager playing video games between 15 and 18 hours a
week?
. In a certain school district, the ages of all new teachers hired during the
last 5 years are normally distributed. Within this curve, 95.4% of the
ages, centered about the mean, are between 24.6 and 37.4 years. Find the
mean age and the standard deviation of the data.
From 1984 to 1995, the winning scores for a golf tournament were 276,
279, 279, 277, 278, 278, 280, 282, 285, 272, 279, and 278. Using the
standard deviation for the sample, Sx, find the percent of these winning
scores that fall within one standard deviation of the mean.
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The mean of a normally distributed set of data is 56, and the standard
deviation is 5. In which interval do approximately 95.4% of all cases lie?
(1) 46­56
(3) 51­61
(2) 46­66
(4) 56­71
In a New York City high school, a survey revealed the mean amount of
cola consumed each week was 12 bottles and the standard deviation was
2.8 bottles. Assuming the survey represents a normal distribution, how
many bottles of cola per week will approximately 68.2% of the students
drink?
(3) 9.2 to 14.8
(1) 6.4 to 12
(4) 12 to 20.4
(2) 6.4 to 17.6
The amount of ketchup dispensed from a machine at Hamburger Palace
is normally distributed with a mean of 0.9 ounce and a standard
deviation of 0.1 ounce. If the machine is used 500 times, approximately
how many times will it be expected to dispense 1 or more ounces of
ketchup?
(1) 5
(3) 80
(2) 16
(4) 100
Mrs. Ramírez is a real estate broker. Last month, the sale prices of homes
in her area approximated a normal distribution with a mean of $150,000
and a standard deviation of $25,000.
A house had a sale price of $175,000. What is the percentile rank of its
sale price, to the nearest whole number? Explain what that percentile
means.
Mrs. Ramírez told a customer that most of the houses sold last month had
selling prices between $125,000 and $175,000. Explain why she is correct.
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The amount of juice dispensed from a machine is normally distributed
with a mean of 10.50 ounces and a standard deviation of 0.75 ounce.
Which interval represents the amount of juice dispensed about 68.2% of
the time?
(3) 9.75­11.25
(1) 9.00­12.00
(4) 10.50­11.25
(2) 9.75­10.50
Twenty high school students took an examination and received the
following scores:
70, 60, 75, 68, 85, 86, 78, 72, 82, 88, 88, 73, 74, 79, 86, 82, 90, 92, 93, 73
Determine what percent of the students scored within one standard
deviation of the mean. Do the results of the examination approximate a
normal distribution? Justify your answer.
The national mean for verbal scores on an exam was 428 and the
standard deviation was 113. Approximately what percent of those
taking this test had verbal scores between 315 and 541?
(3) 38.2%
(1) 68.2%
(4) 26.4%
(2) 52.8%
On a standardized test, the distribution of scores is normal, the mean of
the scores is 75, and the standard deviation is 5.8. If a student scored 83,
the student’s score ranks
(1) below the 75th percentile
(2) between the 75th percentile and the 84th percentile
(3) between the 84th percentile and the 97th percentile
(4) above the 97th percentile
Battery lifetime is normally distributed for large samples. The mean
lifetime is 500 days and the standard deviation is 61 days.
Approximately what percent of batteries have lifetimes longer than 561
days?
(3) 68%
(1) 16%
(4) 84%
(2) 34%
The mean score on a normally distributed exam is 42 with a
standard deviation of 12.1. Which score would be expected to occur
less than 5% of the time?
(3) 60
(1) 25
(4) 67
(2) 32
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During a particular month, a local company surveyed all its employees to determine their travel times to work, in minutes. The data for all 15 employees are shown below.
25 55 40 65 29
45 59 35 25 37
52 30 8 40 55
Determine the number of employees whose travel time is within one standard deviation of the
mean.
Assume that the ages of first­year college students are normally distributed with a mean of 19 years and standard deviation of 1 year. To the nearest integer, find the percentage of first­year college students who are between the ages of 18 years and 20 years, inclusive. To the nearest integer, find the percentage of first­year college students who are 20 years old or older.
In a study of 82 video game players, the researchers found that the ages of these players were normally distributed, with a mean age of 17 years and a standard deviation of 3 years. Determine if there were 15 video game players in this study over the age of 20. Justify your answer.
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