6.1-6.2 Stats notes PDF

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This document is for those who are taking Math 109. The notes are very thorough and easy to understand. ...

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6.1 ●

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Probability model: a description of how a statistician thinks data are produce. ○ When the word model to remind that our description does not really explain how the data came into existence. ○ Example: if a model says that the probability of getting heads when we flip a coin is 0.54, but in fact we get heads 50% of the time, we suspect that the model is not a good match. Probability distribution (Probability Distribution Function; pdf): a tool that helps us by keeping track of the outcomes of a random experiment and the probabilities associated with those outcomes. ○ Example: a playlist that has 10 songs: 6 are Rock, 2 are Country, 1 HipHop, and 1 Opera. Put the playlist on shuffle. Outcome

Probability

Rock

6/10

Non-Rock

4/10

KEYPOINT: A probability distribution tells us (1) all the possible outcomes of a random experiment, and (2) the probability of each outcome. ● Discrete Outcomes (or discrete variables): numerical values that you can list or count. ○ Example: the number of phone numbers stored on the phones of your classmates. ● Continuous Outcomes (or continuous variables): cannot be listed or counted because they occur over a range. ○ Example: the length of your next phone call will last a continuous variable. Example 1: Discrete or Continuous Consider these variables: A. The weight of a submarine sandwich you’re served at a deli. C  ontinuous B. The elapsed time from when you left your house to when you arrived in class this morning. Continuous C. The number of people in the next passing car. Discrete D. The blood-alcohol level of a driver pulled over by the police in a random sobriety check. (Blood-alcohol level is measured as the percent of the blood that is alcohol). Continuous E. The number of eggs laid by a randomly selected salmon as observed in a fishery. Discrete

Discrete Probability Distributions Can Be Tables or Graphs

Example: UCLA’s statistics class was approx. 40% male and 60% female. Using the arbitrarily code the males as 0 and females as 1. By selecting at random, what is the probability that a person is female?

Female

Probability

0

0.40

1

0.60

Discrete Distributions Can Also Be Equations Example: A married couple decide to keep having children until they have a girl. How many children will they have, assuming that boys and girls are equally likely and that the gender of one birth doesn’t depend on any of the previous births? -

(½)x

The probability of having x c hildren is Examples:

-

1

The probability that they have 1 child that is a girl is: (½) -

-

4

The probability that they have 4 children is (½) 10

The probability that they have 10 children is: (½)

=

=½

1 16

= 0.00098

Continuous Probabilities Are Represented as Areas under Curves ● ● ●

The Probabilities for a continuous-valued random experiment are represented as areas under curves. The curve is called a probability density curve. The total area under the curve is 1, because this represents the probability that the outcome will be somewhere on the x-axis. The y-axis in a continuous-valued pdf is labeled “Density.”

6.2 ● ●

The Normal model is the most widely used probability model for continuous numerical variables. The Central Limit Theorem links the Normal model to several key statistical ideas.

Visualizing the Normal Distribution ●

Normal Curve/Normal Distribution: the curve drawn on the histogram. ○ Sometimes called the Gaussian distribution

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The mean of a probability distribution sits at the balancing point of the probability distribution. The standard deviation of a probability distribution measures the spread of the distribution by telling us how far away, typically, the values are from the mean.

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Mean of a Probability Distribution: represented by the Greek letter

μ

(mu, pronounced

“mew”) ●

Standard deviation of a Probability Distribution: represented by the character

σ (sigma).

KEYPOINT: Because the Normal distribution is symmetric, the mean is in the exact center of the distribution. The standard deviation determines whether the Normal curve is wide and low (large standard deviation) or narrow and tall (small standard deviation).

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The only way to distinguish among different Normal distribution is by their means and standard deviations. We can take advantage of this fact to write a short-hand notation to represent a particular Normal distribution. ★ The notation N( μ , of

σ ) represents a Normal distribution that is centered at the value

μ (the mean of the distribution) and whose spread is measured by the values of σ

(the standard deviation of the distribution). KEYPOINT: The Normal distribution is symmetric and unimodal (“bell-shaped”). The notation ( μ ,

σ

) tells us the mean and standard deviation of the Normal distribution.

Finding Normal Probabilities KEYPOINT: When you are finding probabilities with Normal models, the first and most helpful step is to sketch the curve, label it appropriately, and shape in the region of interest.

Without Technology: The Standard Normal ●

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Because the mean is 0 standard deviations away from itself, and any point 1 standard deviation away from the mean is 1 standard unit. (To convert the mean and standard deviation to standard units). Standard Normal model: the Normal model with mean 0 and standard deviation 1.

KEYPOINT: N(0,1) is the standard Normal model: a Normal model with a mean of 0 ( μ = 0) and a standard deviation of 1 ( σ = 1).

x−μ σ (Formula to find z-Scores for Normal probabilities) Example: Find the probabilities that a randomly selected women is shorter than 62 inches. This person is selected from a population of women whose heights follow a N(64,3) distribution. 1. Convert 62 inches to standard units. Call this number z 2. Look up the area below z i n the table for the N(0,1) distribution. ★ IMPORTANT: z =

Example 5: Small Pups Suppose the length of a newborn seal pup follows a Normal distribution with a mean length of 29.5 inches and a standard deviation of 1.2 inches. Question: What is the probability that a newborn pup selected at random is shorter than 28.0 inches? Solution: Begin by converting the length 28.0 inches into standard deviation 28 − 29.5 Z = = −1.5 = 1.25 1.2 1.2 Next we sketch the area that represents the probability we wish to find. We want to find the area under the Normal curve and to the left of 28 inches, or, in standard units, to the left of -1.25.

Answer: The probability that a newborn seal pup will be shorter than 28 inches is about 11%. Check Table on bottom of page 268 for more information. Example 6: A Rand of Seal Pup Lengths Suppose that the N(29.5, 1.2) model is a good description of the distribution of seal pups’ length. Question: What is the probability that this randomly selected seal pup will be between 27 inches and 31 inches long? Solution: To find the area between two values, we break it down into a two step process. 1. We find the area less than 31 inches 2. Then we “chop off” the area below 26 inches The area that remains is the region between 27 and 31 inches. To find the area less than 31 inches, we convert 31 inches to standard units:

(31 − 29.5) = 1.25 1.2 The probability for 31 inches is 0.8944 Z =

To find the area below 27 inches, you must: (27 − 29.5) Z = = - 2.08 1.2 The probability for 27 inches is 0.0188

Then, we subtract (or “chop off”) the smaller area from the big one: 0.8944 - 0.0188 = 0.8756 Answer: The probability that a newborn seal pup will be between 27 and 31 inches long is about 88%. Finding Measurements from Percentiles for the Normal Distribution Percentile:  we are given a probability, but we want to find the value that corresponds to the probability. ★ To find the percentile without technology, you’ll first need to use the standard Normal curve, N( 0,1), to find the z - score from the percentile. Then you must convert the z-score to the proper units. Finding a measurement from a percentile is a two-step process: 1. Find the z-score from the percentile 2. Convert the z-score to the proper units Percentiles of Normal distributions are also known as “finding inverse Normal values.” Example 8: Finding Measurements from Percentiles by Hand Assume that women’s heights follow a Normal distribution with mean 64 inches and standard deviation 3 inches: N( 64, 3). Question: Using the Normal table in Appendix A, confirm that the 25th percentile height is 62 inches. Solution: 1. Find the z-score from the percentile. To do this, you must first find the probability within the table. For the 25th percentile, use a probability of 0.25.

You can now see that the z-score corresponding to the a probability of 0.2514 is -0.67. 2. Convert the z-score to the proper units. A z-score of -0.67 tells us that this value is 0.67 standard deviations below the mean. We need to convert this to a height in inches. One standard deviation is 3 inches, so 0.67 standard deviation is: 0.67 x 3 = 2.0 inches The height is 2.0 inches below the mean. The mean is 64.0 inches, so the 25th percentile is: 64.0 - 2.0 = 62.0 Answer: The women’s height at the 25th percentile is 62 inches, assuming that women’s heights follow a N ( 64, 3) distribution.

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The Normal Model It is a model of a distribution for some numerical variables.

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It provides us with a model of the distributions of probabilities for many real-life numerical variables. - HOW? T he probabilities are represented by the area underneath the bell-shaped curve. - HOW IS IT USED? If the Normal model is appropriate, it can be used for finding probabilities or for finding measurements associated with particular percentiles....