Smoothing the histogram: The Normal Curve (Chapter 8)
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1 Sept. 3 Statistic for the day: Compared to same month the previous year, average monthly drop in U.S. traffic fatalities from May 27 to Feb. 28:.2% Drop in March 28: 22.1% Drop in April 28: 17.9% Smoothing the histogram: The Normal Curve (Chapter 8) A histogram tends to be rough. To replace it with a bell shaped curve: Center the bell at the mean. The bell should be just wide enough so that the middle 95% of the bell is standard deviations wide. This makes systematic, accurate predictions of all sorts possible, provided the bell shape is appropriate for the underlying population. Recall this dataset on handspans from the previous lecture: Men (n = 78) Histogram of HandSpan, with Normal Curve Women (n = 89) HandSpan Mean = 2.86 Standard deviation = Histogram of HandSpan, with Normal Curve Histogram of, with Normal Curve HandSpan Mean = 2.86 Standard deviation = Mean = 68 inches or 5 feet 8 inches Standard deviation = inches 1
2 Research Question 1: If I built my doors 75 inches (6 feet 3 inches) high, what percent of the people would have to duck? Histogram of, with Normal Curve Research Question 2: How high should I build my doorways so that 99% of the people will not have to duck? 6 7 Question 1 (x=75) 8 Question 2 (x=??) Q1: The value of x is 75; find the amount of distribution above it. Q2: Find the value of x so that 99% of the distribution is below it. Z-Scores: Measurement in Standard Deviations 75 mean Z = = = 1.75 SD Compute your Z-score. 1. How many standard deviations are you above or below the mean. Use: Mean = 68 inches Standard deviation = inches 2. Now use the table from the book (p. 157) to determine what percentile you are. Compare s of Females and Males 8 7 Stat 1 students Sp1 Assume male heights have a normal distribution with mean 7 inches and st dev 3 inches. Assume female heights have a normal distribution with mean 6 inches and st dev 3 inches. What is your Z-Score within your sex? What is your percentile within your sex? 6 Female Sex Male 2
3 Answer to Question 1: What percent of people would have to duck if I built my doors 75 inches high? 3 Histogram of, with Normal Curve Recall: 75 has a Z-score of 1.75 From the standard normal table in the book:.96 or 96% of the distribution is below Hence,. or % is above % in here So % of the distribution is above 75 inches Question 1 (x=75) The value at x is 75; find the amount of distribution above it. Convert 75 to Z = 1.75 and use Table 8.1 on p Question 2: What is the value so that 99% of the distribution is below it? (called the 99 th percentile.) 1. Look up the Z-score that corresponds to the 99 th percentile. From the table: Z = Now convert it over to inches: = h h 99 = () = 77.3 Therefore, 99% of the distribution is shorter than 77.3 inches (6 foot 5.3 inches) and that s how high the door should be built Histogram of, with Normal Curve 6 99% in here 7 8 Question inches is the 99th percentile To find the value so that 99% of the distribution is below it: Look up the Z-score for the 99 th percentile and convert it back to inches. Answer these questions: To answer question 1, first convert 72 inches to a z-score: (Assume that adults heights are normally distributed with mean 68 inches and standard deviation inches.).8 or 8% 3
4 Answer to Question 2: What is the first quartile of heights? Translation: First quartile means 25 th percentile, which means.25 are below that height. From p. 157: Find the z-score corresponding to the 25 th percentile..67 Now convert this z-score into a height: h 68 Z score = h = 68 + ( Z score) Shaquille O Neal is 7 feet 1 inch or 85 inches tall. How many people in the country are taller? We will assume that heights are normally distributed with mean 68 inches and standard deviation inches. O Neal s Z-score is Z = (85-68)/ =.25. In other words O Neal is.25 standard deviations above the mean(!) There is only.11 of the normal distribution above.25 standard deviations. There are roughly 35 million people in US. About 9% are over the age of 2 (Census Bureau). That is about 15 million. Page 157 Hence, there should be roughly.11 times 15 million or 165 people taller than Shaquille O Neal. Note: This is an extremely rough calculation, since the normal distribution approximation is less accurate at the extremes. Also, cutting off at age 2 might miss some tall teens! Page 158 Suppose someone claims to have tossed a fair coin 1 times and got 7 heads. Would you believe them?
5 Toss a coin 1 times Repeat 5 times and form a histogram Number of heads So the distribution of the number of heads in 1 tosses of a fair coin is: 1. What is the mean? 2. What is the standard deviation? 3. Let s suppose the smooth version is normal. 5
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