Introduction to Normal Distribution

Show your work for each problem.

1.
What shape does the normal distribution have?
1. a)J-shaped (skewed right)
2. b)U-shaped
3. c)Bell-shaped (symmetric)
4. d)Uniform (flat)
Answer: Bell-shaped (symmetric)
The normal distribution is bell-shaped and symmetric around the mean. Most values cluster near the center, with fewer values in the tails.
2.
In a normal distribution, which percentage of data falls within 1 standard deviation of the mean?
1. a)68%
2. b)95%
3. c)99.7%
4. d)50%
Answer: 68%
According to the 68-95-99.7 rule, approximately 68% of data in a normal distribution falls within 1 standard deviation of the mean ().
3.
In a normal distribution, what percentage of data falls within 2 standard deviations of the mean?
Answer: 95
According to the 68-95-99.7 rule, approximately 95% of data falls within 2 standard deviations of the mean ().
4.
In a standard normal distribution, what are the values of the mean () and standard deviation ()?
1. a),
2. b),
3. c),
4. d),
Answer: ,
The standard normal distribution has mean (centered at zero) and standard deviation (unit spread). This allows us to compare values from any normal distribution using z-scores.
5.
A student scores 75 on a test with mean and standard deviation . What is the z-score?
Answer: 1
. The student scored 1 standard deviation above the mean.
6.
A data point has value in a distribution with and . What is the z-score?
Answer: -1.5
. The negative z-score indicates the value is 1.5 standard deviations below the mean.
7.
Heights are normally distributed with cm and cm. What percentage of people are between 158 cm and 182 cm?
1. a)68%
2. b)95%
3. c)50%
4. d)99.7%
Answer: 95%
and . This is standard deviations, so approximately 95% of people fall in this range.
8.
Test scores are normally distributed with and . Find the z-score for a student who scored 680.
Answer: 1.8
1. What is the formula for z-score? Z = (X - mu) / sigma
2. What is (the difference from the mean)? 180
3. What is ? 1.8
9.
Weights of packages are normally distributed with kg and kg. If a package has a z-score of , what is its weight?
Answer: 42
1. What formula converts z-score back to raw value? X = mu + Z * sigma
2. What is (the deviation)? -8
3. What is ? 42
10.
A z-score of means the value is:
1. a)1.5 standard deviations below the mean
2. b)1.5 times the mean
3. c)1.5 standard deviations above the mean
4. d)At the 1.5th percentile
Answer: 1.5 standard deviations above the mean
A z-score tells us how many standard deviations a value is from the mean. Since 1.5 is positive, the value is 1.5 standard deviations ABOVE the mean.
11.
Battery life is normally distributed with hours and hours. What battery life corresponds to the 84th percentile (z-score )?
Answer: 54
1. What z-score corresponds to the 84th percentile? 1
2. Using , what is ? 6
3. What is ? 54
12.
In a normal distribution with and , what value has a z-score of ?
Answer: 170
. This value is 1.2 standard deviations below the mean.
13.
IQ scores are normally distributed with and . What range of IQ scores contains the middle 68% of the population?
Answer: 85 to 115
1. For the middle 68%, how many standard deviations from the mean? 1
2. What is the lower bound: ? 85
3. What is the upper bound: ? 115
14.
In a normal distribution, what percentage of data is MORE than 2 standard deviations above the mean?
1. a)2.5%
2. b)2.15%
3. c)5%
4. d)16%
Answer: 2.5%
Since 95% of data is within , that leaves 5% outside. Due to symmetry, half (2.5%) is above and half (2.5%) is below .
15.
Exam scores are normally distributed with and . Student A scores 88 and Student B scores 64. Who performed better relative to the class, and by how much?
Answer: Student A by 3 standard deviations
1. Calculate Student A's z-score: 2
2. Calculate Student B's z-score: -1
3. What is the difference in z-scores (how much better is A)? 3