statistical reasoning for everyday life

Statistical Reasoningfor everyday life

Intro to Probability and Statistics

Mr. Spering – Room 113

6.1 & 6.2 Significance and Basic Probability

Is it Normal? Heights of 5,000 randomly selected sunflowers… YES--NORMAL The percentage of Coke six packs that contain six cans… 100%--UNIFORM Sums of the means of rolling 1000 dice… YES--NORMAL A specially designed circuit with only an output of 110 volts… NO--UNIFORM Percentage of students with an IQ higher than 68 when the

mean IQ score is 100 and σ = 16… 97.5%--NORMAL


STATISTICAL SIGNIFICANCEA set of measurements or observations in a statistical study is said to be statistically significant if it is unlikely to have occurred by chance.

Example: A detective in Detroit finds that 25 out of 62 guns used in crimes during the past week were sold by the same gun shop. Is this significant?

Definitely! This is significant because, it is more than likely that there are many gun shops in the Detroit area, if 25 out of 62 guns come from the same shop, in the same week… CMS?? It’s very unlikely to have occurred by chance. {CMS---Common Mathematical Sense}


Quantifying Statistical Significance… What is probability?

Probability is the likelihood of an event occurring in terms of its statistical significance…MORE MATHEMATICAL CONNECTIONS TO PROBABILITY WILL FOLLOW.

How can we quantify statistical significance and CMS {common mathematical sense}?

If the probability of an observed difference occurring by chance is 0.05 (or 1 in 20) or less, the difference is statistically significant at the 0.05 level. (In other words if the difference between the actual and observable variable is less than 5% then it is unlikely to have

occurred by chance at the 0.05 level.)

1

20


Quantifying Statistical Significance…If the probability of an observed difference occurring by chance is 0.01 (or 1 in 100) or less, the difference is statistically significant at the 0.01 level. (In other words if the difference between the actual and observable variable is less than 1% then it is unlikely to have occurred by chance at the 0.01 level.)

Remember ---there is no guarantee for statistical significance. Correlation never implies causation!

1

100


Quantifying Statistical Significance… EXAMPLE:

In the test of the Salk polio vaccine, 33 of 200,000 children in the treatment group got polio, while 115 of the 200,000 in the control group got polio. Based on the 0.01 significance level, is the difference statistically significant.

ANSWER: Based on the large sample, the percent of chance is relatively high, however, the difference in cases must be less than 0.01 and optimistically it is statistically significant. Recall all significance is dependent on significance level, when in doubt use CMS.

{Common

Mathematical Sense}


Quantifying Statistical Significance… EXAMPLE:

In a test on herbal cold remedies, 30 treatment subjects out of 100 contracted a cold. Also, 32 control subjects out of 100 contracted a cold. Based on the 0.01 significance level, is the difference statistically significant.

ANSWER: Based on the sample of 100, the percent of chance is relatively high, however, the difference in cases must be less than 0.01 and hence it this particular case it is not statistically significant. Recall all significance is dependent on significance level, when in doubt use CMS.

{Common

Mathematical Sense}

6.1 & 6.2 Significance and Basic Probability QUESTION & ANSWER TIME …Is the difference between

occurred and chance value significant?

1. In 100 coin tosses, you observe 30 tails. Significant.2. In 100 rolls of a number cube, a 3 appears 16 times. Not Significant3. In 100 rolls of a number cube, a 2 appears 28 times. Significant4. The first 20 cars you see during a trip are convertibles. Significant5. An 85% free throw shooter hits 24 out of 30 free throws. Not Significant

6.1 & 6.2 Significance and Basic Probability QUESTION & ANSWER TIME …Is the difference between occurred

and chance value significant?

6. It rains for 5 days in a row in Phoenix in August. Significant.7. 40 students who attended a study session averaged a 78.9% on

an exam, while 40 students who did not attend a study session averaged a 77.3% on an exam.

Not Significant8. If we assume the mean body temperature is 98.6, and the

probability of finding a sample mean of 98.2 is 0.000000001. Significant9. The first 20 cars you see during a trip are Turquoise. Significant10. A baseball team with a win/loss average of 0.650 wins 12 out of

20 games. Not Significant

6.1 & 6.2 Significance and Basic Probability PROBABILITY: (Theoretical Method) The probability of an event, expressed P(event), is always

between 0 and 1 inclusive. A probability of 0 means that the event is impossible, and a probability of 1 means that the event is certain. Outcome – basic result of an observation or measurement Event – a collection of one or more outcomes having the same property of interest.

1. Check all outcomes are equally likely

2. Count the total number of possible outcomes

3. Count the number of ways the event of interest, A, can occur. Probability Of An Event

P(A) = The Number Of Ways Event A Can Occur The Total Number Of Possible Outcomes


COUNTING OUTCOMES… Suppose process A has a

possible outcomes, process B has b possible outcomes, and process C has c possible outcomes. Assuming the outcomes of the processes do not affect each other, the number of different outcomes for all of the processes combined is a×b×c. {Tree diagram and counting principle}

2 2 2 8

Possibilities of tossing three coins

Possibilities of tossing three coins

8222


PROBABILITY: (Relative Frequency Method) Empirical probability, relative frequency, or experimental

probability, is the ratio of the number favorable outcomes to the total number of trials, not in a sample space but in an actual sequence of experiments. In a more general sense, empirical probability estimates probabilities from experience and observation.

1. Repeat or observe a process many times and count the number of times the event of interest, A, occurs.

2. Estimate P(A) = number of times A occurs

total number of observations


PROBABILITY: (Subjective Method) A subjective probability describes an individual's personal judgment about

how likely a particular event is to occur. It is not based on any precise computation but is often a reasonable assessment by a knowledgeable person.

Like all probabilities, a subjective probability is conventionally expressed on a scale from 0 to 1; a rare event has a subjective probability close to 0, a very common event has a subjective probability close to 1.

A person's subjective probability of an event describes his/her degree of belief in the event

Estimate P(A) = “percentage of belief based on experience”

PRACTICE???


PROBABILITY: Approaches? Identify the probability method.

The chance that you will get married in the next year. Subjective Method, this is dependent on how you feel now. The chance of passing away in an automobile accident is 1 in

7,000. Relative Frequency Method, based on past DOMV data. The chance of rolling an even number on a number cube is 0.5. Theoretical Method, based on theoretical possible outcomes


Complement of an event… The complement of an event, A, expressed as ,

consists of all outcomes in which A does not occur. The probability of is given by

Ex. In a grocery store the scanning system was successful 384 out of 419. What is the probability that the scanner will not work?

A

A

( ) 1 ( )P A P A

384( ) 1 0.084 8.4% chance

419P A


Probability Distribution…

Is a table or graph or formula that gives the probability of all events. It has two properties:

1. Each probability must be a number between 0 and 1 inclusive.

2. The probabilities must total to 1. (Area under normal curve is 100%)

0 0.5 1

Probability


HOMEWORK: Pg 236 # 1-10, 12, 14, 17, 21, 23 Pg 247 # 1-21 omit #12,

and #29, 40

statistical reasoning for everyday life

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