• Density curve– Always on or above horizontal axis– Area under curve equal to 1

• Symmetric density curves have equal mean and median

• Normal distribution1. Mean=Median2. Symmetric, unimodal3. Area under curve = 1 (100%)

The Normal Distribution

Mean and spread of the normal distribution

Figure 1.28Introduction to the Practice of Statistics, Sixth Edition

© 2009 W.H. Freeman and Company

Density curves with the same mean but different standard deviations.

Standard deviation =0.5

Standard deviation =1.0Standard deviation =1.5

• Approximately 68% of the ordered data will fall within one standard deviation of the mean

• Approximately 95% of the ordered data will fall within two standard deviations of the mean

• Approximately 99.7% of the ordered data will fall within three standard deviations of the mean

Empirical Rule(68-95-99.7% Rule)

Empirical Rule: 68-95-99.7% Rule

Figure 1.29Introduction to the Practice of Statistics, Sixth Edition

© 2009 W.H. Freeman and Company

Empirical Rule

34%34%

13.5%

2.35%

13.5%

2.35%0.15%0.15%

How many Standard Deviations away from the mean

1. The average high temperature for the month of April is 65˚F with a standard deviation of 5˚F . Between what values do 68% of April temperatures fall?

A. 60 to 70B. 55 to 75C. 70 to 80D. 90 to 100

EXAMPLES:

2. The average high temperature for the month of April is 65˚F with a standard deviation of 5˚F . How high are the highest 2.5% of temperatures for the month of April?

A. 75 and higherB. 70 and higherC. 65 and higherD. 80 and higher

3. The average high temperature for the month of April is 65˚F with a standard deviation of 5˚F . 99.7% of the temperatures fall into what range?

A. 70 to 80B. 60 to 70C. 50 to 80D. 90 to 100

4. In the summer, a grocery store brings in a large supply of watermelons. The mean weight in pounds is 22. The standard deviation is 4. What percent of watermelons weigh less than 18 pounds?

A. 34%B. 16%C. 2.5%D. 68%

5. In the summer, a grocery store brings in a large supply of watermelons. The mean weight in pounds is 22. The standard deviation is 4. What percent of watermelons weigh more than 30 pounds?

A. 34%B. 16%C. 2.5%D. 68%

6. In the summer, a grocery store brings in a large supply of watermelons. The mean weight in pounds is 22. The standard deviation is 4. What percent of watermelons weigh between 18 and 30 pounds?

A. 34%B. 81.5%C. 95%D. 68%

a.)The middle 68% of apples weigh between _____ and _____.

b.)The middle 95% of apples weigh between _____ and _____.

c.)The middle 99.7% of apples weigh between _____ and _____.

d.) Approximately what percent of apples weigh below 6oz?e.) Approximately what percent of apples weigh above 4 oz?

7. Weights of apples are normally distributed with a mean of 10 oz and a standard deviation of 2 oz.

• Describes how many standard deviations an observation is from the mean.– Negative z-scores (observation is below the mean)– Positive z-scores (observation is above the mean)– z-score equal to zero (observation is equal to the mean)

– Standardizes any “score”

Z-scores

• If we assume the distribution of the variable is normal, then the z-scores have a standard normal distribution.

Z-scores

s

)-(x z•

x

)-(x

z•

Examples

1. Find z-score for an apple that weighs 11 oz.

2. 15 oz?

3. 5 oz?

4. The average high temperature for the month of April is 65˚F with a standard deviation of 5˚F Find the standard score of an April high temperature of 71˚F.

A. 1.2B. 3.5C. 2.4D. 5

• The standard normal distribution has a mean of 0 and a standard deviation of 1.

• Can use Table A (z-table) to get area under the curve for a standard normal.

• Area under curve = proportion (percent)• Proportions represent probabilities.Examples: (Use the table)• What percent of apples weigh below 7 oz?• What percent of apples weigh more than 5oz?

Standard Normal Distribution

Percentiles

• The cth percentile of a distribution is a value such that c percent of the observations lie below it and the rest lie above.

Example

• What percentage of April high temperatures fall below 71˚F ?

Example

• The length of human pregnancies from conception to birth varies according to a distribution that is approximately normal with mean 266 days and standard deviation 16 days. Use this information to answer the questions below. – Between what values do the lengths of the middle 99.7% of all

pregnancies fall?– What percent of these pregnancies last more than 290 days?– What percent of these pregnancies last between 258 and 290 days?– How long is a pregnancy which falls into the 13.57 percentile?

Example

• Suppose that the average height for adult males is normally distributed with a mean of 70 inches and a standard deviation of 2.5 inches.– What percentile does a man who is 68 inches fall into?– What percent of men are taller than 72 inches?– How tall is a man in the 9.68 percentile?– How tall is a man who has 8% of all men taller than him?– Determine the percentage of men falling between 69.25

inches and 73.5 inches.

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Margin of Error (E or moe)

• z* = is a critical value– 90% z = 1.645– 95% z = 1.96– 99% z = 2.576

• If you know a particular confidence level (%) and MOE, you can solve for your sample size, n.

*margin of error zn

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Margin of Error (E or moe)

• A smaller moe says that we have pinned down the parameter quite precisely.

• To make the margin of error smaller…– make z* smaller– make n bigger, which will

cost more

*margin of error zn