Understanding Standard Normal Distribution and Statistical Analysis
Understanding the Standard Normal Curve
Finding the Area Under the Standard Normal Curve
1. Find the area under the standard normal curve to the left of z = -1.32 (Go to the z-table and find the corresponding value).
2. Find the area under the standard normal curve to the left of z = 1.49, P(Z < 1.49) (Use the z-table, find the value corresponding to the positive z-score, and draw a line to the left).
3. Find the area under the standard normal curve between z = -1.65 and z = 1.65, P(-1.65 < Z < 1.65) (Find the values in the z-table and subtract the smaller value from the larger one).
4. Find the area under the standard normal curve to the right of z = -1.49, P(Z > -1.49) (Find the value in the z-table and subtract it from 1).
Finding Z-Scores
1. Find the z-score that is associated with the 85th percentile (Look for 0.8500 in the z-table; the corresponding z-score is approximately 1.04).
2. If the area under the standard normal curve to the left of a z-score is 0.7088, find the z-score (Subtract 0.7088 from 1 to get 0.2912. Find 0.2912 in the z-table and the corresponding z-score is the answer).
Applications of Normal Distribution
1. A population is approximately normally distributed with a mean of 68 and a standard deviation of 5 points. Find P(65 < X < 75)
- When x = 65, calculate the z-score.
- When x = 75, calculate the z-score.
- Find the corresponding values in the z-table and subtract them.
The sentence will be: If we picked one person at random from this population, the probability that he/she gets a score between 65 and 75 is [calculated probability].
2. What percentage of women from this population would you expect to be more than 75 inches tall? (Calculate the z-score for 75, find the corresponding value in the z-table, and subtract it from 1).
3. When given “less than,” e.g., P(X < 5) (Calculate the z-score and find the corresponding value in the z-table).
4. Suppose you want to find out the grade you need to be in the top 15% of your class on an exam. From past experience, your teacher estimates the mean will be 80 and the standard deviation will be 12. What will be the minimum score needed to be in the top 15% of your class?
Look for 0.8500 (representing the 85th percentile, as the top 15% corresponds to the 85th percentile and above) in the z-table. The corresponding z-score is approximately 1.04.
5. What is the 44th percentile when μ = 89 and σ = 5?
(Find 0.4400 in the z-table and perform the same calculations as above).
Sampling Distribution
A population has a mean of 54 with a standard deviation of 5 and a sample size of 25.
1. The mean value of the sampling distribution would be: μx = 54
2. The standard deviation or standard error of the mean for the sampling distribution will be:
3. What shape would you expect the distribution of all these sample means to have? Normal Distribution
A normal population with μ = 50 and σ = 15. A sample of size 9 is selected at random.
1. Find P(45 < X < 60) (Apply the formula, find the resulting z-score in the z-table, and subtract the values).
2. Find P(X < [value]) (Find the corresponding value in the z-table).
To perform these calculations in a calculator, first perform the operations below, then input x – 1 * (result), and press enter.
3. Find the probability that, if one jar is selected at random, it will contain between [value 1] and [value 2] (Use the formula from Chapter 6).
4. Find the probability that a random sample of 36 jars will have a mean between [value 1] and [value 2] (Use the formula from Chapter 7).
The probability we will randomly choose [scenario] is [calculated probability].
Key Concepts
1. In a standard normal distribution, the mean is 0 and the standard deviation is 1.
2. As sample size increases, the standard error of the mean gets smaller.
3. A randomly chosen (Chapter 7)
4. A random sample of (Chapter 7)
5. One randomly selected pair of batteries
Linear Correlation Analysis
1. A linear relationship exists because we can visualize a line. It seems to be [positive/negative] because of the [+/-] slope.
2. Sometimes there may be a relationship between two variables, but it might not be a linear relationship.
3. The primary purpose of linear correlation analysis is to measure the strength of the linear relationship between two variables.
4. If your linear correlation coefficient is calculated to be 0, it means there is no *linear* relationship between your two variables, not necessarily no relationship at all.
5. If someone says that the linear correlation coefficient is 1.14, you should suspect a calculation error because it cannot be higher than 1 or lower than -1.
6. If you change the independent and dependent variables, the Linear Correlation Coefficient (LCC) would *not* change.
Interpreting Linear Regression
y = 59.1 + 3.631x
1. How would you interpret 59.1? For every increase or decrease of 1 unit in x, the predicted y value will increase or decrease by the slope’s value.
2. What is the y-intercept? It is 3.631, and it means that when x = 0, our predicted y value will be 59.1.
3. Independent variable: x-axis
4. Dependent variable: y-axis
5. Using the equation of the line of best fit, if you have 272 miles of track, what is the predicted number of vehicles you will need?
y = -59.1 + 3.631 * 272
6. The limitations of the line of best fit: It should be used to make predictions only about the population from which the sample was drawn.
7. It has to be within the domain and use current data.
8. Write a sentence to interpret the y-intercept of the equation for the line of best fit.
When a person is 0 inches tall, the predicted weight will be [y-intercept value].
9. Write a sentence interpreting the slope.
For every increase or decrease of 1 inch in height, the predicted weight will increase or decrease by [slope value].
10. Could we use the line of best fit to estimate the weight of a person who is 18 inches tall?
No, because it is outside our domain.
Common Misconceptions
Which of these sentences are false?
- There is a strong linear relationship between gender and height because we found a correlation of 0.5.
- Plant height and leaf height were found to be negatively correlated because the correlation coefficient is -1.41.
- Since the correlation between x and y is 0, this means there is no relationship whatsoever between these two variables.
- All of the above.
Shape of Sampling Distribution
What would the shape of this sampling distribution of sample means be and why?
It will be a normal distribution because the sample size is greater than 30.
True Statements about Random Variables
Which of the following statements is true where x is a random variable with mean μ and standard deviation σ, and x̄ is the sample mean corresponding to a random sample of size n?
- A. If x has a normal distribution, then x̄ has a distribution only if n > 30.
- B. If x has a distribution that is not normal, then x̄ can never have a normal distribution.
- C. If x has a normal distribution, then x̄ has a distribution with mean equal to μ for any sample size n.
True Statements about Sampling Distribution
Which of the following statements are true?
- The sampling distribution of x̄ is normal if the population has a normal distribution.
- When n is large, the sampling distribution of x̄ is approximately normal even if the population is not normally distributed.
- As the sample size increases, the standard error of the mean decreases.
- The dispersion of the sampling distribution of sample means will always be greater than the dispersion of the population. (FALSE)