Estimating Population Variance

Estimating a Population Variance When developing estimates of population variance or standard deviation, we use the chi-square distribution. Chi-Square Distribution is given by: 𝝌 𝟐 = 𝒏−𝟏 𝒔 𝟐 𝝈 𝟐 Where 𝑛 = number of sample values 𝑠 2 = sample variance 𝜎 2 = population variance

Estimating a Population Variance Properties of the Chi-Square Distribution The chi-square distribution is not symmetric.

Estimating a Population Variance Properties of the Chi-Square Distribution The chi-square distribution is not symmetric. The values of the chi-square distribution can only be zero or positive.

Estimating a Population Variance Properties of the Chi-Square Distribution The chi-square distribution is not symmetric. The values of the chi-square distribution can only be zero or positive. The chi-square distribution is different for each number of degrees of freedom, and the number of degrees of freedom df =𝑛−1.

Estimating a Population Variance Properties of the Chi-Square Distribution The chi-square distribution is not symmetric. The values of the chi-square distribution can only be zero or positive. The chi-square distribution is different for each number of degrees of freedom, and the number of degrees of freedom df =𝑛−1. Bad news is since the chi-square distribution isn’t symmetric we have to calculate the lower and upper confidence interval limits separately. Good news is there is no margin of error formula.

Estimating a Population Variance Finding Critical values of 𝝌 𝟐 More Bad news our calculators will not return the critical values for 𝝌 𝟐 . We use a 𝝌 𝟐 table to find the critical values we need know the desired confidence level and the degrees of freedom. Lets find the critical values corresponding to a 95% confidence level and sample size of 12.

Estimating a Population Variance Estimators of 𝝈 𝟐 Recall that 𝑠 2 was an unbiased estimator of 𝜎 2 , so values of 𝑠 2 tend to target 𝜎 2 .

Estimating a Population Variance Estimators of 𝝈 𝟐 Recall that 𝑠 2 was an unbiased estimator of 𝜎 2 , so values of 𝑠 2 tend to target 𝜎 2 . The sample variance 𝒔 𝟐 is the best point estimate of the population variance 𝝈 𝟐 .

Estimating a Population Variance Estimators of 𝝈 𝟐 Recall that 𝑠 2 was an unbiased estimator of 𝜎 2 , so values of 𝑠 2 tend to target 𝜎 2 . The sample variance 𝒔 𝟐 is the best point estimate of the population variance 𝝈 𝟐 . Also recall that sample standard deviation s was not an unbiased estimator of population standard deviation 𝜎. However if the sample size is large the bias is small so…

Estimating a Population Variance Estimators of 𝝈 𝟐 Recall that 𝑠 2 was an unbiased estimator of 𝜎 2 , so values of 𝑠 2 tend to target 𝜎 2 . The sample variance 𝒔 𝟐 is the best point estimate of the population variance 𝝈 𝟐 . Also recall that sample standard deviation s was not an unbiased estimator of population standard deviation 𝜎. However if the sample size is large the bias is small so… The sample standard deviation 𝒔 is the commonly used as a point estimate of 𝝈 (even though it is a biased estimate).

Estimating a Population Variance Constructing a confidence interval for 𝝈or 𝝈 𝟐 Verify that the requirements are satisfied (simple random sample, population must be normally distributed).

Estimating a Population Variance Constructing a confidence interval for 𝝈or 𝝈 𝟐 Verify that the requirements are satisfied (simple random sample, population must be normally distributed). Using 𝑛−1degrees of freedom and excel find the critical values 𝜒 𝑅 2 and 𝜒 𝐿 2 that correspond to the desired confidence level.

Estimating a Population Variance Constructing a confidence interval for 𝝈or 𝝈 𝟐 Verify that the requirements are satisfied (simple random sample, population must be normally distributed). Using 𝑛−1degrees of freedom use table A-4 to find the critical values 𝜒 𝑅 2 and 𝜒 𝐿 2 that correspond to the desired confidence level. Evaluate the upper and lower confidence interval limits using this format of the confidence interval: (𝑛−1) 𝑠 2 𝜒 𝑅 2 < 𝜎 2 < (𝑛−1) 𝑠 2 𝜒 𝐿 2

Estimating a Population Variance Constructing a confidence interval for 𝝈or 𝝈 𝟐 Verify that the requirements are satisfied (simple random sample, population must be normally distributed). Using 𝑛−1degrees of freedom use table A-4 the critical values 𝜒 𝑅 2 and 𝜒 𝐿 2 that correspond to the desired confidence level. Evaluate the upper and lower confidence interval limits using this format of the confidence interval: (𝑛−1) 𝑠 2 𝜒 𝑅 2 < 𝜎 2 < (𝑛−1) 𝑠 2 𝜒 𝐿 2 If a confidence interval estimate of 𝜎 is desired, take the square root of the upper and lower confidence interval limits and change 𝜎 2 to 𝜎.

Estimating a Population Variance Constructing a confidence interval for 𝝈or 𝝈 𝟐 Verify that the requirements are satisfied (simple random sample, population must be normally distributed). Using 𝑛−1degrees of freedom use table A-4 the critical values 𝜒 𝑅 2 and 𝜒 𝐿 2 that correspond to the desired confidence level. Evaluate the upper and lower confidence interval limits using this format of the confidence interval: (𝑛−1) 𝑠 2 𝜒 𝑅 2 < 𝜎 2 < (𝑛−1) 𝑠 2 𝜒 𝐿 2 If a confidence interval estimate of 𝜎 is desired, take the square root of the upper and lower confidence interval limits and change 𝜎 2 to 𝜎. Round, If original data then round to one more decimal place than the original data. If using the sample variance or standard deviation, then round to the same number of decimal places.

Estimating a Population Variance Use the following information to construct a confidence interval for the population standard deviation 𝜎. Speeds of Drivers ticketed in a 65mi/h zone. 95% C-Level; 𝑛=25, 𝑥 =81.0 𝑚𝑝ℎ, 𝑠=2.3 𝑚𝑝ℎ Reaction times of NASCAR Drivers 99% C-level; 𝑛=8, 𝑥 =1.24 𝑠𝑒𝑐, 𝑠=0.12 𝑠𝑒𝑐

Estimating a Population Variance The listed values are waiting times (in minutes) of customers at the Jefferson Valley Bank, where customers enter a single waiting line that feeds three teller windows. Construct a 95% confidence interval for the population standard deviation 𝜎. 6.5 6.6 6.7 6.8 7.1 7.3 7.4 7.7 7.7 7.7

Homework!! 7-5: 1-11 odd, 17-23 odd.