Probability Distributions Statistics Chapter 4 Normal Distribution Distribution

Probability Distributions Statistics Chapter 4

Normal Distribution

Distribution The description of the possible values of a random variable and the possible occurrences of these values.

Normal Distribution Curve A symmetrical curve that shows the highest frequency in the center with an identical curve on either side of that center. Usually called a “bell curve” from its shape.

Discrete Values Data where a finite number of values exist between any two other values. Data points are not joined on a graph.

Continuous Variables that take on any value within the limits of the variable.

Example of Variables If we are counting from 1 to 10 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 would be discrete values. 3. 5 would be a continuous variable.

Other Examples of discrete values: Number of people in class, number of correct answers on a test Examples of continuous values: Lengths, temperatures, ages, heights Can you think of two other examples of each?

Example 1 Does the histogram represent normal distribution?

Example 2 Does the histogram represent normal distribution?

Example 3 Does the histogram represent normal distribution?

Probabilities

Binomial Distributions

Binomial Experiments that involve only 2 choices as answers. (either a success or a failure) Binomial Distribution of the observations in a binomial experiment

Example 4 Keith took a poll of the students in his school to see if they agreed with a new “no cell phone” policy. He found the following results. Make a histogram to show this data. Age Number Responding ‘No’ Is this data normally distributed? 14 56 15 65 16 90 17 95 18 60

Example 5 Jake sent out the same survey as Keith, except he sent it to every junior high school and high school in the district. He found these results. Make a histogram to show this data. Is the data normally distributed? Age Number Respondi ng ‘No’ 12 274 13 261 14 259 15 289 16 233 17 225 18 253 19 245 20 216

Example 6 A local food chain has determined that 40% of the people who shop in the store use an incentive card, such as air miles. If 10 people walk into the store, what is the probability that AT MOST half of these will be using an incentive card?

Example 7 Karen and Denny want to have 5 children after they get married. What is the probability that they will have exactly 3 girls?

TECHNOLOGY When using exactly: Use Binompdf function 2 nd VARS binompdf n value, probability of success, number of successes

TECHNOLOGY When using at least, more than, less than, or at most: Use Binompdf function 2 nd VARS binomcdf n value, probability of success, number of successes

Example 8 A fair coin is tossed 50 times. What is the probability that you will get heads in 30 of these tosses?

Example 9 A fair coin is tossed 50 times. What is the probability that you will get heads in at most 30 of these tosses?

Example 9 A fair coin is tossed 50 times. What is the probability that you will get heads in at least 30 of these tosses?

Exponential Distributions

Exponential Distribution

Three Types of Distribution What are some similarities and differences between the three distribution curves?

Example 10 ABC Computer Company is doing a quality control check on their newest core chip. They randomly chose 25 chips from a batch of 200 to test and examined them to see how long they would continuously run before failing. The following results were obtained.

Example 10 Continued What kind of data is represented in the table? How do we know? Number of Chips Hours to Failure 8 1, 000 6 2, 000 4 3, 000 3 4, 000 2 5, 000 2 6, 000

Example 10 Continued TECHNOLOGY 1. Enter the data STAT Edit Enter data into L 1 and L 2 2. Turn on stat plots 2 nd Y = 1 enter Turn on stat plots Clear 3. Find values STAT CALC EXPREG

Example 10 Continued TECHNOLOGY MAKE SURE: 2 nd 0 Scroll down to DIAGNOSTICON

TECHNOLOGY Coefficient of Determination (r^2): A standard quantitative measure of best fit. Has values from 0 to 1, and the closer the value is to 1, the better the fit is.

Example 11 Radioactive substance are measure using a Geiger-Muller counter. Robert was working in his lab measuring the count rate of a radioactive particle. He obtained the following data.

Example 11 Continued Is this data representative of an exponential distribution? If so find the equation. What would a count be at 7. 5 hours? Time (hr) Count (atoms) 15 544 12 272 9 136 6 68 3 34 1 17