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• 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
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Mean and spread of the normal distribution
Figure 1.28Introduction to the Practice of Statistics, Sixth Edition
© 2009 W.H. Freeman and Company
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Density curves with the same mean but different standard deviations.
Standard deviation =0.5
Standard deviation =1.0Standard deviation =1.5
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• 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)
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Empirical Rule: 68-95-99.7% Rule
Figure 1.29Introduction to the Practice of Statistics, Sixth Edition
© 2009 W.H. Freeman and Company
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Empirical Rule
34%34%
13.5%
2.35%
13.5%
2.35%0.15%0.15%
How many Standard Deviations away from the mean
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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:
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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
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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
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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%
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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%
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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%
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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.
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• 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
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• 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•
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Examples
1. Find z-score for an apple that weighs 11 oz.
2. 15 oz?
3. 5 oz?
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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
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• 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
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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.
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Example
• What percentage of April high temperatures fall below 71˚F ?
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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?
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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