stat 110 chap 13

mean of a density curv

"balancing point" of the density curve if it were solid. • For a symmetric density curve, both the mean and the median will be the same (exactly at center) • For a skewed density curve, the mean will be farther out in the long tail than the median.

The length of horse pregnancies from conception to birth varies according to a roughly Normal distribution with a mean of 336 days and standard deviation 3 days. Use the 68-95-99.7 rule to answer the following question. Fill in the blank: About ______ percent of horse pregnancies are less than 333 days.

16

Congressman Floyd commissions a sample survey of voters to learn what percentage favor him in his race for reelection. To avoid spending too much, he samples only 50 voters. Suppose that only 43 percent of the voters support Floyd. The percentage favoring Floyd in a random sample of size 50 will vary from sample to sample according to a Normal distribution with a mean of 43 percent and a standard deviation of 7 percent. What percentage of all such samples will (wrongly) show that half or more of the voters favor Floyd?

16%

Consider the Normal distribution with a mean of 111 and a standard deviation of 11 as a description of the IQ test scores of all rural Midwest seventh-grade students. Fill in the blank: The percentage of IQ scores for rural Midwest seventh-graders that are less than 100 is ______.

16%

The distribution of heights of young men is approximately Normal with a mean of 70 inches and a standard deviation of 2.5 inches. Use the 68-95-99.7 rule to answer the following question. Fill in the blank: ____ percent of men are shorter than 65 inches. (Give your answer to one decimal place.)

2.5

The length of human pregnancies from conception to birth varies according to a distribution that is approximately Normal with a mean of 266 days and standard deviation 16 days. Use the 68-95-99.7 rule to answer the following question. Almost all (99.7 percent) pregnancies fall between ____ to ____ days.

218, 314

The length of horse pregnancies from conception to birth varies according to a roughly Normal distribution with a mean of 336 days and standard deviation 3 days. Use the 68-95-99.7 rule to answer the following question. The correct range that contains the middle 95 percent of all horse pregnancies is _____ to ______.

330, 342

The distribution of the heights of young men is approximately normal with a mean of 70 inches and a standard deviation of 2.5 inches. The heights of the middle 95 percent of men fall between ___ inches and ___ inches.

65, 75

• We can completely characterize a normal distribution by

giving its mean and standard deviation.

density curve

is a graphical picture of the population distribution of a variable. smooth curve.

• The median of a density curve

is the value with half the area to the left of it and half the area to the right. ("equal-areas point")

Inflection point =

the place where the curve changes from "opening down" to "opening up"

The standard deviation describes

the spread of the distribution.

the sampling distributions of sample proportions and sample means are approximately normal distributions, when the sample size is large.

• Caution: Some data (examples: income data, house price data) tend to be skewed and therefore not normal.

The distribution of the heights of young men is approximately normal with a mean of 70 inches and a standard deviation of 2.5 inches. Fill in the blank: The standard score of a height of 72 inches (6 feet) is ___. (Give your answer to one decimal place.)

.8

The 68-95-99.7 Rule-Empirical Rule.

For a normal distribution, 68% of the data fall within one standard deviation of the mean. • For a normal distribution, 95% of the data fall within two standard deviations of the mean. • For a normal distribution, 99.7% (almost all!) of the data fall within three standard deviations of the mean.

Standard Scores

For an observation from a normal distribution, its standard score is found by subtracting off the mean and then dividing by the standard deviation of the distribution.

The length of human pregnancies from conception to birth varies according to a distribution that is approximately Normal with a mean of 266 days and standard deviation 16 days. Use the 68-95-99.7 rule to answer the following question. How long are the longest 2.5 percent of all pregnancies?

Longer than 298 days

• Example: Suppose SAT Math scores are normally distributed with a mean of 500 and a standard deviation of 100. • Suppose ACT Math scores are normally distributed with a mean of 18 and a standard deviation of 6. • Sarah got a 550 on the SAT math exam. • Bill got a 15 and Julie got a 24 on the ACT math exam.

Sarah's standard score was (550 - 500) / 100 = 0.5, so Sarah was 0.5 standard deviations above the mean. • Bill's standard score was (15 - 18) / 6 = -0.5, so Bill was 0.5 standard deviations below the mean. • Julie's standard score was (24 - 18) / 6 = 1.0, so Julie was 1 standard deviation above the mean.

• Standard score = (observation - mean) / sd

This tells us how many standard deviations above the mean or below the mean an observation is. • This allows us to compare values coming from different distributions.

Density curves always deal with proportions of data, but the plotting is done a bit differently.

Total area under a density curve is always 1 (Total proportion = 1).

first quartile

of a density curve is the value with 0.25 area under the curve to the left of it.

third quartile

of a density curve is the value with 0.75 area under the curve to the left of it.

percentile

of a distribution is a value such that a given percentage of the data lies below that value. The median is the 50th percentile of a distribution, because 50% of the data lie below the median.

The Wechsler Adult Intelligence Scale (WAIS) is an IQ test. The scores on the WAIS for the 20 to 34 years age group are approximately normally distributed with a mean of 110 and a standard deviation of 15. The scores for the 60 to 64 years age group are approximately normally distributed with a mean of 90 and a standard deviation of 15. Sarah, who is 30, scores 130 on the WAIS. Her mother, who is 60, takes the test and scores 115. True or false: Sarah's mother scored higher than Sarah relative to her age group.

t

The mean describes

the center of the distribution.

For a normal density curve

• For a normal density curve, the standard deviation is the distance between the center and the "inflection point"