Ch 6 - Lecture notes 6 PDF

Preview

Summary

Chapter 6: Probability & Normal Distributions...

Description

Desiree Curcio

Chapter 6: Probability & Normal Distributions Key Terms: Normal Distribution- a theoretical distribution with data that are symmetrically distributed around the mean, the median, & the mode.

Y= Standard normal distribution (z-distribution)- a normal distribution w/ a mean equal to 0 & a standard deviation equal to 1; it is distributed in z-score units along the x-axis Z-score- a value on the x-axis of a standard normal distribution. The numerical value of a z-score specifies the distance or standard deviation of a value from the mean. Standard normal transformation (z-transformation)- a formula used to convert any normal distribution w/ any mean & any variance to a standard normal distribution w/ a mean equal to 0 & a SD equal to 1. z = x –μ for population of scores σ z=x–M for sample of scores SD Unit normal table (z-table)- a type of probability distribution table displaying a list of zscores & the corresponding probabilities (or proportions of area) associated w/ each zscore listed. Real limits- of the outcome of a binomial variable are the upper & lower values within which the probability of obtaining that outcome is contained. The real limits for a binomial outcome, x, is x ± 0.5 Steps for: Locating proportions: Step 1- Transform raw score (x) into a z-score Step 2- Locate the corresponding proportion for the z-score in the unit normal table Locating scores: Step 1- Locate a z-score associated w/ a given proportion in the unit normal table Step 2- Transform the z-score into a raw score (x)

To use the normal approximation:

Step 1- Check for normality Step 2- Compute the mean & SD Step 3- Find the real limits Step 4- Locate the z-score for each real limit Step 5- Find the proportion located within real limits

End-of-Chapter Problems: Factual Problems 2. It is applied to behavioral research because in physical, behavioral, and social measurement studies things tend to be normal. 4. Mean= 0; SD= 1 6. The score minus the mean, divided by the standard deviation 8. Step 1- Locate a z-score associated w/ a given proportion in the unit normal table Step 2- Transform the z-score into a raw score (x) 10. 10

Concepts & Application Problems 12. a) .1915+.5000= .6915 b) .0934 c) .5000 d) .0250 e) .4602 14. a) equal b) first is bigger c) first is bigger d) second is bigger e) second is bigger

16. a) Top 5%  .0500 z = 1.645

b) Bot. 2.5%  .0250 z = -1.96 c) Top 69.5% .3050 z= -.51 d) Top 50% .5000 z = 0 e) Bot. 50%  .5000 z = 0 18. M= 3.30; SD= 0.20 1.28= x-3.3 .2 .256= x-3.3 3.556= x GPA must be at least a 3.56 20. M= 23; SD= 2.5 a) z= (19-23)/2.5= -1.6 p= .0548 b) z= (25.5-23)/2.5= 1 p= .1587 c) z= (19-23)/2.5= -1.6 p= .4452 d) z= (25.5-23)/2.5= 1 p= .8413 e) z= (19-23)/2.5= -1.6 p= .9452

22. M= 10; x= 12.47; p= .0500; z= 1.645 1.645= 12.47-10 = 2.47 SD SD 1.645 = 1 2.47 SD SD= 2.47 = 1.50 1.645 24. SD= 10; x= 10; p=.5000; z= 0 M= 10

Problems in Research

26. Since the data are normally distributed people who score very highly will be towards the tail of the curve meaning that there won’t be many scores that are high. People will be somewhere in the middle of very high and very low scores. The low and high scores will be less frequent and towards the end. 28. Good: 93.8 ± 2.9; Average: 84.8 ± 3.0 a) z= (90-93.8)/2.9= -1.31 p= .9049 b) z= (90-84.8)/3= 1.73 p=.0418 30. Hydrolysate: 64.8 ± 9.6; Milk-based: 101 ± 12.1 a) z= (86.8-64.8)/9.6=2.29 p= .0110 z= (101-64.8)/9.6= 3.77 p= .0001 p in between= .0110 b) z= (86.8-101)/12.1= -1.17 p= .1210 z= (101-101)/12.1= 0 p= .5000 p in between= .3790 32. M= 14.93; SD= 5.65 a) p= .1000 z= 1.28 1.28= x-14.93 5.65 7.23= x-14.93 22.16= x b) z= 10-14.93 = -.87 5.65 p= .3078+.5000= .8078 np= 86(.8078)= 69.5...