HW3 - E. Johnson - Graded
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Feb 20, 2024
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EHS 2550
Basic Statistics for Health Sciences
Homework No 3 - Normal Distribution and Central Limit Theorem
Name:
Emma Johnson
Date:
Grade: 27.4/30 = 91.3% Good work!
1. Label the following normal distribution (
x
= 30, s = 5). Label the mean, 1, 2, and 3 standard deviations above and
below the mean. Enter a number or text in the proper box below (3 points). 2.9/3
2. Label the following standard normal distribution
. Label the mean, 1, 2, and 3 standard deviations above and below the mean. Enter a number or text in the proper box below (3 points). 2.9/3
3. The scores of the Medical College Admission Test (MCAT) for the year 2022 showed an average score of 125.9 in the section “Psychological, Social, and Biological Foundations of Behavior” and a standard deviation of 3. Suppose you scored a 129.1 in this section. Your standardized score was calculated to be 1.07. Using the z-tables (Find the z-
tables in the folder name “Useful documents and references” in the General tab in Moodle), determine the percent of the population of medical school candidates who took the MCAT that scored higher than you in this section. Show your work (2 points). 2/2
Answer:
According to the Z table at a score of 1.07, we did better than 85.77% of people which means 14.23%
did better than us. 1
35
30
25
1
0
-1
EHS 2550
Basic Statistics for Health Sciences
Homework No 3 - Normal Distribution and Central Limit Theorem
4. Below is the normal distribution of heights of American men. If a man is 68.2 inches tall, what percentage of the population is he taller than? What percentage is he shorter than? Hint: Use the z-tables. Show your work (3 points). 1/3
Answer:
Z=68.2-70/3= -.06
According to the Z table the man is taller than 47.6% which means 52.4% of the population is taller than the man. Please note that (68.2-70)/3 = -0.6 (and not z = -0.06). The probability value corresponding to z=-
0.6 is 0.2743. Therefore, the referenced man was taller than 27.43 % of American men and shorter than 75.27% (1 – 0.2743) of American men.
Answer Questions 5 and 6 based on the following frequency curve, which represents the normal distribution of the
water bill prices (in USD) for a family of four based on each person using the national average of 100 gallons per day.
5. Pretend your water bill is $97.68. How many standard deviations is this value above the mean? Hint: Use the z-
tables and show your work (1 point). 0.9/1
2
88
73
58
Answer:
Z= 97.68-73/15= 1.65
We are less than 2 full deviations away from the mean, we are 1.65 deviations from the mean.
The correct answer is 1.645 standard deviations above the mean. Please remember the following when rounding numbers: when the number to round is followed by a 5 it remains even or becomes even. So, 1.645 is rounded down to 1.64. Of course, if the number 4 was followed by a 6 or greater, you round up to 1.65. Similarly, if the number 4 was followed by 4 or less, you round down to 1.64.
EHS 2550
Basic Statistics for Health Sciences
Homework No 3 - Normal Distribution and Central Limit Theorem
6. Calculate the z-score for a water bill at $106.30. What percentage of water bills are higher than that? Hint: Use the z-tables and show your work (3 points). 3/3
Answer:
Z=106.30-73/15=2.22
The water bill is 98.7% higher than the populations therefore 1.3% pay higher than 106.3
7. Label the following frequency curve using the Empirical Rule. Enter in the boxes where 68%, 95%, and 99.7% of the data lie (3 points). 3/3
8.
Historically, the final grade scores for EHS 2550 given in the Winter semester are normally distributed with a mean of 2.9 and standard deviation of 0.6. What percentage of students who took EHS 2550 in the Winter had
a final grade between 2.3 and 3.5? Hint: Use the empirical rule and show your work (2 points). 2/2
Answer:
2.9-2.3=.6 or 1 standard deviation below the mean
2.9+.6=3.5 or 1 standard deviation above the mean
Therefore. 68% of the students scored between 2.3 and 3.5
9.
The lifespans of gorillas in a particular zoo are normally distributed. The average gorilla lives 20.8 years; the standard deviation is 3.1 years. Use the empirical rule, estimate the probability of a gorilla living less than 23.9 years. Show your work (2 points). 2/2
Answer:
20.8+3.1=23.9 or 1 standard deviation above the mean
34+34+13.5+2.4+.1=84
Therefore there is an 84% chance the gorilla will live less than 23.9 years
3
99
.7
%
95
%
68
%
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EHS 2550
Basic Statistics for Health Sciences
Homework No 3 - Normal Distribution and Central Limit Theorem
10.
A person scores 3 standard deviations above the mean on a test (x = 60, s = 5). What percentage of the class had a higher score? Show your work (3 points). 2.8/3
Answer:
Z=75-60/5=3
3.00=99.87
100-99.87=.13 – This number is already a percentage. 0.13% scored higher. This is the exact answer.
1.3% scored higher.
Please note that according to the empirical rule 3 standard deviations below and above the mean encompass 99.7 % of the data. Then, 100 – 99.7 = 0.3% is left on both sides. Therefore, only 0.3%/2
= 0.15% is above 3 standard deviations above the mean.
11.
Two samples are taken from a population. The first sample has n=25 and the second sample has n=500. Which
sample is more likely to have a bell-shaped curve? Explain your answer. (2 points). 1.9/2
Answer:
The sample with 500 participants has a higher likelihood of being a bell-shaped curve because the more people in the sample, the more representative it is of the population. The perfect answer would have also mentioned the Central Limit Theorem
12.
How many of the following values are within one standard deviation of the mean? 180, 313, 101, 255, 202, 198, 109, 183, 181, 113, 171, 165, 318, 145, 131, 145, 226, 113, 268, 108. Hint: Calculate the average and the standard deviation first. Show your work (3 points). 3/3
Answer:
180+313+ 101+ 255+ 202+ 198+ 109+ 183+ 181+ 113+ 171+ 165+ 318+ 145+ 131+ 145+ 226+ 113+ 268+ 108/20= 181.25
S=66.3
11 are within one standard deviation from the mean.
I wish you could have listed the values that are within 1 standard deviation below and above the mean.
Your answers are correct but you did not show your work. When a question asks you to show your work, it is for me to see that you can do the calculations. If
the final answer is not correct, I can still give points for the correct formula, the correct substitution, etc.
During the final exam, mind you, I will not ask students to show their work. It is expected that they
should know how to select the proper formula, how to calculate, etc.. The only thing they’ll need to enter is the final answer. If it’s not correct, then it’s all or nothing. 4
EHS 2550
Basic Statistics for Health Sciences
Homework No 3 - Normal Distribution and Central Limit Theorem
Total points: 30 5