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5.0 Continuous Probability Distributions
5.1 Introduction
5.1.1 Name and sketch the graph of the various probability distributions
The following are examples of some of the different types of probability distributions.
Continuous Probability Distributions:
1) The normal distribution: (This is the one we will be dealing with)
2) The uniform distribution:
3) The Exponential distribution:
5.2.1/5.2.2 Discuss and sketch the normal distribution and state the properties of the normal distribution
The Normal Distribution Curve:(Bell-Curve)
A population which is normally distributed will have a mean located at the center and a curve that is symmetrical (which means that each side is a reflection of each other)The percentages given below stem from the Empircal Rule which states that 68.26% of the data lie within one standard deviation of the mean; 95.46% of the data lie within two standard deviations of the mean and 99.73% of the data lie within three standard deviations from the mean.
The normal distribution is a continuous probability distribution that is uniquely determined by its mean (µ ) and standard deviation (σ ).
A normal distribution is completely described by its mean and standard deviation. This indicates that if the mean and standard deviation are known, a normal distribution can be constructed and its curve drawn.
The following chart shows three normal distributions, where the means are the same but the standard deviations are different.
The following chart shows three distributions with different means but identical standard deviations.
The following chart shows three distributions with different means and different standard deviations.
Examples:
1. A company conducts a test on the lifespan of a battery. For a particular battery, the mean life is 19 hours. The useful life of the battery follows a normal distribution with a standard deviation of 1.2 hours. Answer the following questions.
a. About 68% of the batteries will have a lifespan between what two values?
b. About 95% of the batteries will have a lifespan between what two values?
c. Virtually all of the batteries will have a lifespan between what two values?
2. The mean of a normal probability distribution is 250; the standard deviation is 20.
a. About 95% percent of the observations lie between what values?
b. About what percent of the data lies between 230 and 270?
c. About what percent of the data lies between 190 and 310?
5.3 The Standard Normal Curve
2.4.4.4 and 2.4.4.5 Define and compute z-scores.
The Standard Normal Distribution:
This is a normal distribution curve but in a standard normal distribution the mean is zero and the standard deviation is 1.
Remember: the standard deviation is a measurement of how much a particular value deviates from the mean.
The standard normal distribution can be used for all problems where the normal distribution is applicable.Any normal distribution can be converted into the "standard normal distribution" by using a z value.The z value measures the distance between a particular value of X and the mean in units of the standard deviation.
This is how to compute a z-score:
where:
X: value of your random variable
: the mean of the distribution of the random variable
: the standard deviation of the distribution
(So, looking at the formula you can see that the z-score measures how many standard deviations a number is from the mean.)
The following illustration demonstrates converting an X value to a standardized z value:
A comparison of z-values and standard deviations is shown in the following illustration.
Examples:
1. A distribution has a mean of 100 and a standard deviation of 10. Calculate the z-scores of each of the following.
a. 110
b. 80
2. The weekly incomes of shift foremen in the glass industry are normally distributed with a mean of $1000 and a standard deviation of $100. What is the z-value for a foreman who earns:
a. $1150 per week?
b. $925 per week?
5.3.1 Use the standard normal curve to calculate probabilities.
You will be using another table(Appendix D) to calculate probabilities using the standard normal curve. It contains a list of z-scores.
Obtaining the Probability:(2 STEPS!)
1. To obtain the probability of a value falling in the interval between the variable of interest (X) and the mean, we first compute the z-score.
2. To obtain the probability we refer to the Standard Normal Probability Table (Appendix D) for the associated probability of a given area under the curve. The following is an illustration of how we read the Standard Normal Probability Table for z = 0.12. (SKETCHES HELP!!!)
Note: The table gives the probability for the area under the curve from the mean to the z-value.And remember the distribution is symmetrical (50% of values are on the right and 50% are on the left)
Examples:
1. A normal population has a mean of 1000 and a standard deviation of 100.
a. Compute the z-value associated with 1000. (although you don’t really need to for this one since it’s the mean!)
b. Compute the z-value associated with 1100.
b. What is the probability of selecting a value between 1000 and 1100?
d. What is the probability of selecting a value that is less than 1100?
e. What is the probability of selecting a value that is greater than 1100?
2. A recent study of the hourly wages of a group of employees showed that the mean hourly salary was $20.50, with a standard deviation of $3.50. If we select a crew member at random, what is the probability the crew member earns:
a. Between $20.50 and $24.50 per hour?
b. More than $24.50 per hour?
c. Less than $24.50 per hour?
d. Less than $19.00 per hour?
e. more than $19.00 per hour?
*f. between $19.00 and $24.50?
***g. between $22.50 and $24.50?
Worksheet for 5.0
1. The mean of a normal probability distribution is 500; the standard deviation is 10.
a. About 68 percent of the observations lie between what values?
b. About 95 percent of the observations lie between what two values?
c. Practically all of the observations lie between what two values?
2. The mean of a normal probability distribution is 60; the standard deviation is 5.
a. About what percent of the observations lie between 55 and 65?
b. About what percent of the observations lie between 50 and 70?
c. About what percent of the observations lie between 45 and 75?
3. The Kamp family has twins, Rob and Rachel. Both Rob and Rachel graduated college 2 years ago, and each is now earning $50 000 per year. Rachel works in the retail industry, where the mean salary for executives with less than 5 years’ experience is$35000 with a standard deviation of $8 000. Rob is an engineer. The mean salary for engineers with less than 5 years’ experience is $60 000 with a standard deviation of $5000. Compute the z values for both Rob and Rachel.
4. A recent article reported that the mean labour cost to repair a heat pump is $90 with a standard deviation of $22. A company completed repairs on two heat pumps this morning. The labour cost for the first was $75 and it was $100 for the second. Compute z values for each and comment on your findings.
5. A normal population has a mean of 20.0 and a standard deviation of 4.0.
a. Compute the z value associated with 25.0.
b. What proportion of the population is between 20.0 and 25.0?
c. What proportion of the population is less than 18.0?
6. A normal population has a mean of 12.2 and a standard deviation of 2.5.
a. Compute the z value associated with 14.3.
b. What proportion of the population is between 12.2 and 14.3?
c. What proportion of the population is less than 10.0?
7. A recent study of the hourly wages of maintenance crew members for majo airlines showed that the mean hourly salary was $20.50, with a standard deviation of $3.50. If we select a crew member at random, what is the probability the crew member earns:
a. between $20.50 and $24.00 per hour?
b. more than $24.00 per hour?
c. less than $19.00 per hour?
8. The mean of a normal distribution is 400 pounds. The standard deviation is 10 pounds.
a. What is the area between 415 pounds and the mean of 400 pounds?
b. What is the area between the mean and 395?
c. What is the probability of selecting a value at random and discovering that it has a value of less than 395 pounds?