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What does a population that is normally distributed look like

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**1. **What does a population that is normally distributed look like?

**2. **Empirical Rule Empirical Rule is sometimes referred to as the 68-95-99.7% Rule. Again, to use the Empirical Rule the distribution of the data must be normal bell-shaped curve.Empirical Rule is sometimes referred to as the 68-95-99.7% Rule. Again, to use the Empirical Rule the distribution of the data must be normal bell-shaped curve.

**3. **Empirical Rule—restated 68% of the data values fall within 1 standard deviation of the mean in either direction
95% of the data values fall within 2 standard deviation of the mean in either direction
99.7% of the data values fall within 3 standard deviation of the mean in either direction
Remember values in a data set must appear to be a normal bell-shaped histogram, dotplot, or stemplot to use the Empirical Rule!
Rule of Thumb: Range divide by 6 approximately the standard deviation.Rule of Thumb: Range divide by 6 approximately the standard deviation.

**4. **Empirical Rule Take time to slowly click through slide.
Stress that the Empirical Rule can ONLY be used with the assumption that the distribution is normal (bell-shaped curve). Sixty-eight percent of the ordered data of a normal distribution lies within one standard deviation of the mean. Ninety-five percent of the ordered data of a normal distribution lies within two standard deviations of the mean. And, 99.7% of the ordered data of a normal distribution lies within 3 standard deviations of the mean. The normal distribution here, in this example shown, has a mean of 0 and standard deviation of 1.Take time to slowly click through slide.
Stress that the Empirical Rule can ONLY be used with the assumption that the distribution is normal (bell-shaped curve). Sixty-eight percent of the ordered data of a normal distribution lies within one standard deviation of the mean. Ninety-five percent of the ordered data of a normal distribution lies within two standard deviations of the mean. And, 99.7% of the ordered data of a normal distribution lies within 3 standard deviations of the mean. The normal distribution here, in this example shown, has a mean of 0 and standard deviation of 1.

**5. **Average American adult male height is 69 inches (5’ 9”) tall with a standard deviation of 2.5 inches.

**6. **Empirical Rule-- Let H~N(69, 2.5) Let H~N(69, 2.5) means H represents the variable height for an adult male has a normal distribution with population mean of 69 inches and a population standard deviation of 2.5 (or variance of 2.5).
Start with an easy problem using the empirical rule AND introduce proper probability notation. Explain P implies probability (likelihood of an event), H represents the random variable of height of adult male, P(h<69) is the question in statistical notation.Let H~N(69, 2.5) means H represents the variable height for an adult male has a normal distribution with population mean of 69 inches and a population standard deviation of 2.5 (or variance of 2.5).
Start with an easy problem using the empirical rule AND introduce proper probability notation. Explain P implies probability (likelihood of an event), H represents the random variable of height of adult male, P(h<69) is the question in statistical notation.

**7. **Using the Empirical Rule Let H~N(69, 2.5) means H represents the variable height for an adult male has a normal distribution with population mean of 69 inches and a population standard deviation of 2.5 (or variance of 2.5).
Let H~N(69, 2.5) means H represents the variable height for an adult male has a normal distribution with population mean of 69 inches and a population standard deviation of 2.5 (or variance of 2.5).

**8. **Using Empirical Rule-- Let H~N(69, 2.5) There are several ways to do this problem. Some students may see it as .5 - .34 to obtain .16.
If students obtained an answer of .16 but did it differently please ask them to discuss their method.There are several ways to do this problem. Some students may see it as .5 - .34 to obtain .16.
If students obtained an answer of .16 but did it differently please ask them to discuss their method.

**9. **Using Empirical Rule--Let H~N(69, 2.5)

**10. **Using Empirical Rule--Let H~N(69, 2.5)

**11. **Z-Scores are standardized standard deviation measurements of how far from the center (mean) a data value falls.

**12. **Standardized Z-Score To get a Z-score, you need to have 3 things
Observed actual data value of random variable x
Population mean, ? also known as expected outcome/value/center
Population standard deviation, ?
Then follow the formula. Inform students that z-scores are used to compare similar information by calculating z-scores.
Also, Z-distribution is the normal distribution with a population mean of 0 and a population standard deviation of 1. Z~N(0, 1).
Inform students that z-scores are used to compare similar information by calculating z-scores.
Also, Z-distribution is the normal distribution with a population mean of 0 and a population standard deviation of 1. Z~N(0, 1).

**13. **Empirical Rule & Z-Score About 68% of data values in a normally distributed data set have z-scores between –1 and 1; approximately 95% of the values have z-scores between –2 and 2; and about 99.7% of the values have z-scores between –3 and 3. Since the Z-distribution is normal, the empirical rule holds.Since the Z-distribution is normal, the empirical rule holds.

**14. **Z-Score & Let H ~ N(69, 2.5) What would be the standardized score for an adult male who stood 71.5 inches? Slowly go through the steps on how to calculate z-score. Remember to explain notation.Slowly go through the steps on how to calculate z-score. Remember to explain notation.

**15. **Z-Score & Let H ~ N(69, 2.5)

**16. **Comparing Z-Scores Suppose Bubba’s score on exam A was 65, where Exam A ~ N(50, 10). And, Bubbette score was an 88 on exam B, where Exam B ~ N(74, 12).
Who outscored who? Use Z-score to compare.

**17. **Comparing Z-Scores Heights for traditional college-age students in the US have means and standard deviations of approximately 70 inches and 3 inches for males and 165.1 cm and 6.35 cm for females. If a male college student were 68 inches tall and a female college student was 160 cm tall, who is relatively shorter in their respected gender groups?

**22. **Using TI-83+

**25. **What if I know the probability that an event will happen, how do I find the corresponding z-score?

**29. **Using TI-83+