The Normal Distribution Areas under the normal ...
Exam #2 Study Guide – SOCL 2201 Chapter 6 – The Normal Distribution
Areas under the normal distribution (68, 95, 99%)
(50% lie above or to the right of the center, and vice versa) 68%: Between the mean and the +1 standard deviation, 68% of all observations occur 95%: Between the mean and the +2 standard deviation, 95% of all observations occur 99%: Between the mean and the +3 standard deviation, 99% of all observations occur
Z scores: the number of standard deviations that a given raw score is above or below the mean The Standard Normal Table (begins on pg. 493)
Turning a raw score into a Z score Score – Mean = Z score
(80-70.7) / 10.27 = 0.97 (z score)
Standard dev.
Z score into a raw score:
Y=Mean + zscore (standard dev) Y= 70.07 + (-1.5)(10.27)= 70.07- 15.41= 54.66 Y= 70.07 + (1.7)(10.27)= 70.07+ 17.46= 87.53
Turning a Z score into a proportion (and vice versa)
85- 70.07 = 1.45 10.27
Find 1.45 in appendix B, 0.4265 corresponds to 1.45 0.4265 x 100 = 42.65%
Finding a raw score or Z score that applies to a percentage / proportion
Percentile higher than 50: Percentile 95 95 / 100 = .95 1.00 - .95 = .05 Appendix B, column C, entry closest to .05 is .0495 which corresponds to a Z score of 1.65. Y= 70.07 + 1.65(10.27) = 87.02
Percentile lower than 50: Percentile 40 40 / 100 = 0.40 In column C closest to 0.40 is 0.4013 which corresponds to a Z score of -0.25. Y= 70.07 + (-0.25)(10.27) = 67.50
Locating percentiles in a normally distributed variable
Score above the mean: 85-70.07 = 1.45 Appendix B, column C, 0.0735 corresponds to 1.45. 10.27
1.00 – 0.0735 = 0.9265 = 92.65 %
Score below the mean: 65-70.07 = -0.49 10.27
Appendix B, Column C, 0.3121 corresponds to .49 100 x 0.3121 = 31.21%
Applying skills used in the last three points to answer practical questions
Chapter 7 – Sampling and Sampling Distributions
The Sampling Distribution – A theoretical probability distribution of all possible sample values for the statistic in which we are interested
The Sampling Distribution of the Mean- a theoretical distribution of sample means that would be obtained by drawing from the population all possible samples of the same size.
The Standard Error of the Mean- The standard deviation of the sampling distribution of the mean. It describes how much dispersion there is in the sampling distribution of the mean. Standard Dev of population = standard error of mean √ sample size
14,687 = 8,480 √3
The Central Limit Theorem- If all possible random samples of size N are drawn from a population with a mean, and a standard dev, then as N becomes larger the sampling distribution of the sample means becomes approximately normal, with mean and standard dev. -
Even when the population distribution is skewed, you can still assume that the sampling distribution of the mean is normal, given random samples of large enough size.
CLT assures us that: -
As the sample size gets larger, the mean of the sampling distribution becomes equal to the population mean
-
As the sample size gets larger, the standard error of the mean decreases in size.
Chapter 8 – Estimation Estimated Standard Error for means Standard dev of population = SE √Number in sample
Estimated Standard Error for proportions SE for proportions = √ ((pop portion)(1- pop portion) Pop size P= .51 N= 1,013
√ ((0.51)(1- 0.51) = 0.016 1,013)
Calculating the appropriate Z score based on the level of confidence 90% = 1.65 95% = 1.96 99% = 2.58 93%= 93 / 2 = .465 (area between mean and Z) 7/2= .035 (area beyond Z) Z=1.81
Confidence Intervals for means - Calculate standard error of the mean SD of population / Square root of N
1.5 / √500 = 0.07
-
Find Z score that corresponds to confidence level ex: 90% = 1.65
-
Calculation: 90% CI = 7.5 ± 1.65(0.07) 7.5 ± 0.12 7.38 to 7.62
Confidence Intervals for proportions P (sample of proportion) = .51 N= 1,013
√ ((0.51)(1- 0.51) = 0.016 1,013) 95% CI = 0.51 ± 1.96(0.016) 0.51 ± 0.03 0.48 to 0.54
Interpreting Confidence Intervals / Confidence Interval error (From above) 95 % confident that the population’s true mean / proportion is somewhere between the interval of 0.48 and 0.54 Error = 5%, if all possible samples were collected, 5% of the intervals calculated would not contain the true mean / proportion
Relationships between sample size / level of confidence / standard deviation and confidence interval width
•
Level of confidence – Increase confidence, wider interval
•
Sample size – Increase sample size, narrower interval
•
Standard deviation – Increase SD, wider interval
•
Opposite relationships apply
Areas under the normal distribution (68, 95, 99%)
(50% lie above or to the right of the center, and vice versa) 68%: Between the mean and the +1 standard deviation, 68% of all observations occur 95%: Between the mean and the +2 standard deviation, 95% of all observations occur 99%: Between the mean and the +3 standard deviation, 99% of all observations occur
Z scores: the number of standard deviations that a given raw score is above or below the mean The Standard Normal Table (begins on pg. 493)
Turning a raw score into a Z score Score – Mean = Z score
(80-70.7) / 10.27 = 0.97 (z score)
Standard dev.
Z score into a raw score:
Y=Mean + zscore (standard dev) Y= 70.07 + (-1.5)(10.27)= 70.07- 15.41= 54.66 Y= 70.07 + (1.7)(10.27)= 70.07+ 17.46= 87.53
Turning a Z score into a proportion (and vice versa)
85- 70.07 = 1.45 10.27
Find 1.45 in appendix B, 0.4265 corresponds to 1.45 0.4265 x 100 = 42.65%
Finding a raw score or Z score that applies to a percentage / proportion
Percentile higher than 50: Percentile 95 95 / 100 = .95 1.00 - .95 = .05 Appendix B, column C, entry closest to .05 is .0495 which corresponds to a Z score of 1.65. Y= 70.07 + 1.65(10.27) = 87.02
Percentile lower than 50: Percentile 40 40 / 100 = 0.40 In column C closest to 0.40 is 0.4013 which corresponds to a Z score of -0.25. Y= 70.07 + (-0.25)(10.27) = 67.50
Locating percentiles in a normally distributed variable
Score above the mean: 85-70.07 = 1.45 Appendix B, column C, 0.0735 corresponds to 1.45. 10.27
1.00 – 0.0735 = 0.9265 = 92.65 %
Score below the mean: 65-70.07 = -0.49 10.27
Appendix B, Column C, 0.3121 corresponds to .49 100 x 0.3121 = 31.21%
Applying skills used in the last three points to answer practical questions
Chapter 7 – Sampling and Sampling Distributions
The Sampling Distribution – A theoretical probability distribution of all possible sample values for the statistic in which we are interested
The Sampling Distribution of the Mean- a theoretical distribution of sample means that would be obtained by drawing from the population all possible samples of the same size.
The Standard Error of the Mean- The standard deviation of the sampling distribution of the mean. It describes how much dispersion there is in the sampling distribution of the mean. Standard Dev of population = standard error of mean √ sample size
14,687 = 8,480 √3
The Central Limit Theorem- If all possible random samples of size N are drawn from a population with a mean, and a standard dev, then as N becomes larger the sampling distribution of the sample means becomes approximately normal, with mean and standard dev. -
Even when the population distribution is skewed, you can still assume that the sampling distribution of the mean is normal, given random samples of large enough size.
CLT assures us that: -
As the sample size gets larger, the mean of the sampling distribution becomes equal to the population mean
-
As the sample size gets larger, the standard error of the mean decreases in size.
Chapter 8 – Estimation Estimated Standard Error for means Standard dev of population = SE √Number in sample
Estimated Standard Error for proportions SE for proportions = √ ((pop portion)(1- pop portion) Pop size P= .51 N= 1,013
√ ((0.51)(1- 0.51) = 0.016 1,013)
Calculating the appropriate Z score based on the level of confidence 90% = 1.65 95% = 1.96 99% = 2.58 93%= 93 / 2 = .465 (area between mean and Z) 7/2= .035 (area beyond Z) Z=1.81
Confidence Intervals for means - Calculate standard error of the mean SD of population / Square root of N
1.5 / √500 = 0.07
-
Find Z score that corresponds to confidence level ex: 90% = 1.65
-
Calculation: 90% CI = 7.5 ± 1.65(0.07) 7.5 ± 0.12 7.38 to 7.62
Confidence Intervals for proportions P (sample of proportion) = .51 N= 1,013
√ ((0.51)(1- 0.51) = 0.016 1,013) 95% CI = 0.51 ± 1.96(0.016) 0.51 ± 0.03 0.48 to 0.54
Interpreting Confidence Intervals / Confidence Interval error (From above) 95 % confident that the population’s true mean / proportion is somewhere between the interval of 0.48 and 0.54 Error = 5%, if all possible samples were collected, 5% of the intervals calculated would not contain the true mean / proportion
Relationships between sample size / level of confidence / standard deviation and confidence interval width
•
Level of confidence – Increase confidence, wider interval
•
Sample size – Increase sample size, narrower interval
•
Standard deviation – Increase SD, wider interval
•
Opposite relationships apply
