University:
Amherst College
Course:
STAT 111 | Introduction to Statistics
Academic year:

2023
Views:

485

Pages:

17
Author:

Finnegan Wolf

Continuous Random Variables ______________________________________ 1) Review the differences between continuous and discrete RVs. 2) Describe the properties of a uniform distribution and the method for calculating the area beneath the curve. 3) Describe the properties of a normal distribution and of the standard normal curve. 4) Calculate the area beneath the SNC of a given interval. 5) Learn to apply the normal approximation to the binomial distribution. Distinguishing Continuous and Discrete RVs ______________________________________ Continuous RVs Freedom to vary: • Discrete RVs Freedom to vary: • • Represented by: • Represented by: • p (x = a): • • p (x = a): • Mathematical description: • Mathematical description: • How do we work with continuous distributions? ______________________________________ We are still interested in the rare event approach. We still want to know the likelihood of an event. P (x=a) = 0, so… • How do we do that? a) b) c) Nothing complicated about a) and c). • Uniform Distribution ______________________________________ 1 / d-c where (c  x  d) (in other words, c and d are the lower and upper boundaries of the distribution)  = c + d / 2 (arithmetic mean)  = d – c /  12 f(x) = p (a  x  b) = Area c-d Area of a rectangle = base * height base = b – a; height = f(x) p (a  x  b) = Area b-a = (b-a)f(x) Properties of a Normal Distribution ______________________________________ 1) Continuous 2) Symmetrical 3) Bell-shaped 4) Area under the curve equals 1 • Y-Axis: relative frequency height is arbitrary ______________________________________ Why do we use it a lot? 1) It has some very nice properties EX: 2) Many things approximate a normal dist. EX: Exceptions: What affects P(a  x  b)? ______________________________________ Height   Problem: Are we going to have to engage in a whole set of complicated calculations (i.e., CALCULUS) every single time we want to find P(a  x  b)? No. How are we going to do that? Standard Normal Curve ______________________________________ Just like any Normally Distributed Variable... 1) Continuous 2) Symmetrical 3) Bell-shaped 4) Area under the curve equals 1 What makes it special... 1)  = 0 2)  = 1 ______________________________________ Why that is good? So what? How do we do that? Steps for calculating the area beneath the SNC ______________________________________ 1) Always draw a picture a) bell-shaped curve b) tick mark for the mean c) tick marks at  1 and 2 SDs (maybe even 3) d) cut-off points at the boundaries of the interval you are interested in e) Shade in the region you are interested in 2) Set up your (in)equation p(a  z  b) or p(z  b) or p(z  b) 3) Consult the Normal Curve Area Table p (area underneath curve between 0 and z). 4) Potentially tricky part use information from table to calculate requested probability Finding the Area Examples ______________________________________ a) What is P(-1.62  z  1.62)? b) What is P(-1.47  z  2.03)? c) What is P(.84  z  1.39)? d) What is P(-.84  z  -1.39)? d) What is P(z  1.16)? f) What is P(z  1.16)? g) What is P(z  -1.16)? h) What is P(z  -1.16)? Simplified Rules for using the Unit Normal Table ___________________________________________ p (A  z  B) If both are negative: If a is neg and b is pos: If both are positive: ______________________________________ P(z  A) If A is negative: If A is positive: ______________________________________ p( z  A) If A is negative: If A is positive: Steps for calculating the probability of an event for any normally distributed variable ______________________________________ 1) Draw the picture • include numbers for actual distribution underneath those for Std. Normal Curve 2) Set up proportion (in)equation • p(a  x  b) 3) Convert to z values using formula • p([a -  / ]  z  [b -  / ]) 4) Use table to calculate proportion 5) Potentially tricky part Not tricky anymore! Using the uhhhhh unit normal uhhhh table: The Uhhhhhhhhhhhh example ______________________________________ Professor Binns has a bad habit of saying “Uhhhhhhhh” over and over throughout his History lectures. Let’s say that the number of “Uhhhhhhs” produced by Professor Binns in a given minute is normally distributed with  = 30 and  = 5. What is the probability that Prof. Binns utters between 22 and 32 “Uhhhhhhs” a minute? P(22  Uhhh  32) P ([22-30 / 5]  z  [32-30 / 5]) P ([-8 / 5]  z  [2 / 5]) P (-1.6  z  .40) According to the table: the area below .40 (Body) = .6554 the area below -1.6 (Tail) = .0548 Therefore, P(22  Uhhh  32) = Mmmmm….Doughnuts ______________________________________ Your good pal Biff has found that he can eat 18 doughnuts in one sitting without becoming ill. He crows that he must be in the 95%-ile of doughnut eaters nationwide. You go home and consult the latest issue of “Doughnut Eating in America” and find that the number of doughnuts that a typical person can eat without becoming ill is normally distributed with a mean of 14 and a standard deviation of 2.25. Find the actual 95%-ile and determine whether Biff’s estimate of his doughnuteating prowess is correct? Conclusion: How tall am I? How tall are you? ______________________________________ 1a) According to the data collected at the beginning of the semester, the average height in this class is ____ inches with a SD of ____. What is P (x  72")? b) If we were to pick one person from the class at random, what is the probability that s/he would be my height or shorter? 2) What is my percentile rank among men if the mean for males is ____ inches with a SD of ____? 3) What would my percentile rank be if I was a woman and the mean height for women is ____ with a SD of ____? Steps for using the normal approximation for a binomial distribution ______________________________________ 1) Determine n and p 2) Calculate  (np) and  (npq) 3) Confirm that   3 is a legal observation 4) Draw the picture • include values of actual distribution 5) Set up (in)equation • be sure to use continuity correction 6) Convert to z values using formula 7) Use table to calculate proportion Girl Power ______________________________________ How many people think the Spice Girls are “totally awesome”? Let’s say we believe that 60 percent of the people like the Spice Girls, but when we ask a group of 50 people we find that only 20 claim to be fans? What is the p (x  20) if we believe that the proportion of people who actually like the Spice Girls is .60? np = .60 * 50 = 30  = np(1-p) = (50)*(.60)*(.40) = 3.5 30  3 * 3.5 = 30  10.5 = [19.5 – 40.5] (Both legal). P(x  20.5), with correction P(z  [20.5-30] / 3.5) P(z  -9.5 / 3.5) P(z  -2.71) Area of the SNC beneath –2.71 = Tail (2.71) = .0034. Conclusion: a) error of estimation b) sampling error Is Handedness genetic? ______________________________________ A researcher is interested in whether handedness is genetic. She knows that 18% of the general population is left-handed. She decides to collect data on the children of left-handed parents to see how likely they are to be lefties, as well. She interviews 1000 children, and finds that 205 are left-handed. What is the probability of obtaining this sample if we expect 18% of the population to be left-handed? np =  = 180  3 *

Description

Related Documents

Free up your schedule!

Take 5 seconds to unlock