ESTIMATION Prepared by: Paolo Lorenzo Bautista

Estimation  We wish to estimate a characteristic of the population, by using information from the sample  Types of Estimates  Point Estimate – single value  Interval Estimate – interval or range Est : PLBautista

ESTIMATION OF PARAMETERS One Population  One Mean  Case 1  Case 2  One Proportion  One Variance Two Populations  Difference of Two Means  Case 1 to 3  Paired Observations  Difference of Two Proportions  Ratio of Two Variances Est : PLBautista

Estimation for One Mean  Best Point Estimate:  Interval Estimation for μ  CASE 1: σ is known, OR n ≥ 30  The (1-α)100% confidence interval for μ is given by MARGIN OF ERROR Est : PLBautista

Example 1  The mean and standard deviation for the sample point average of a random sample of 36 college seniors are calculated to be 2.6 and 0.3 respectively. Find a 95% confidence interval for the grade point average of the entire senior class. Est : PLBautista

Determining the Sample Size  If the sample mean is used as an estimate of µ, we can be (1-α)100% confident that the error will not exceed E when the sample size is  How large a sample do we need if we do not wish our error to exceed 0.05? Est : PLBautista

Case 2: σ is unknown, AND n < 30  A (1-α)100% confidence interval for μ is given by  The t-value has n-1 degrees of freedom MARGIN OF ERROR Est : PLBautista

Example 2  A random sample of 16 mid-sized cars tested for fuel consumption gave a mean of 26.4 miles per gallon with a standard deviation of 2.3 miles per gallon. Assuming that the miles per gallon given by all mid- sized cars have a normal distribution, find a 99% confidence interval for the population mean. Est : PLBautista

Exercise  In an effort to estimate the mean amount spent per customer for dinner at a major Atlantis restaurant, data were collected for a sample of 49 customers.  Assume a population standard deviation of $5, and a 95% confidence level. What is the margin of error?  If the sample mean is $34.80, what is the 95% confidence interval for the population mean? Est : PLBautista

Exercise  In the testing of a new production method, 18 employees were selected randomly and asked to try the new method. The sample mean production rate was 80 pph, and the sample s.d. was 10 pph. Provide 90% and 95% confidence intervals for the population mean production rate for the new method, assuming the population has a normal probability distribution. Est : PLBautista

Estimation for the Proportion of Success  Assume a binomial experiment is being conducted Est : PLBautista

Estimation for the Proportion of Success  Best point estimate:  A (1-α)100% confidence interval for p is given by MARGIN OF ERROR Est : PLBautista

Example 3  In a random sample of n = 500 families owning television sets in Quezon City, it was found that x = 340 subscribed to Destiny. Find a 95% confidence interval for the actual proportion of families in this city who subscribe to Destiny. Est : PLBautista

Determining the Sample Size  If we use the sample proportion as an estimate of p, we can be (1 – α)100% confident that the error will not exceed E when the sample size is approximately  How large a sample would be needed if we do not wish our error to exceed 0.02 at 95% confidence? Est : PLBautista

Exercise  A random sample of 75 college students is selected and 16 are found to have cars on campus.  Use a 95% confidence interval to estimate the fraction of students who have cars on campus.  How large a sample is needed if we do not wish our error to exceed 0.075?  In a random sample of 1000 homes in a certain city, it is found that 628 are heated by natural gas. Find the 98% confidence interval for the fraction of homes in this city that are heated by natural gas. Est : PLBautista

Estimation for the Variance  Recall: Est : PLBautista

Estimation for the Variance  Best Point Estimate:  A (1-α)100% confidence interval for σ 2 is given by  The chi-squared values have n-1 degrees of freedom Est : PLBautista

Example 4  The following are the volumes, in deciliters, of 10 cans of peaches distributed by a certain company: 46.4, 46.1, 45.8, 47.0, 46.1, 45.9, 46.9, 45.8, 45.2, and Find a 95% confidence interval for the variance of all such cans assuming that volume is normally distributed. Est : PLBautista

ESTIMATION OF PARAMETERS One Population  One Mean  Case 1  Case 2  One Proportion  One Variance Two Populations  Difference of Two Means  Case 1 to 3  Paired Observations  Difference of Two Proportions  Ratio of Two Variances Est : PLBautista

Estimation for the Difference of Two Means  Best Point Estimate:  Interval Estimation:  Case 1:  A (1-α)100% confidence interval for μ 1 -μ 2 is given by MARGIN OF ERROR Est : PLBautista

Example 5  A standardized chemistry test was given to 50 girls and 75 boys. The girls made an average grade of 76 with a standard deviation of 6, while the boys made an average grade of 82 with a standard deviation of 8. Find a 96% confidence interval for the difference of the two population means. Est : PLBautista

Case 2: σ 1 =σ 2 unknown, AND n 1 <30 and n 2 <30  A (1-α)100% confidence interval for μ 1 -μ 2 is given by  The t-value has Est : PLBautista

Example 6  A course in statistics is taught to 12 students by the transmissive teaching method. A second group of 10 students was given the same course by means of the transformative teaching method. At the end of the trimester the same examination was given to each group. The 12 students who used the transmissive method got an average grade of 85 with a standard deviation of 4, while the 10 students under the transformative method had an average grade of 81 with a standard deviation of 5. Find a 90% confidence interval for the difference between population means, assuming the populations are approximately normally distributed with equal variances. Est : PLBautista

Case 3: σ 1 ≠σ 2 unknown, AND n 1 <30 and n 2 <30  A (1-α)100% confidence interval for μ 1 -μ 2 is given by  The t-value has degrees of freedom given by: Est : PLBautista

Example 7  Records for the past 15 years have shown the average rainfall in a certain region of the country for the month of May to be 4.93 centimeters, with a standard deviation of 1.14 centimeters. A second region of the country has had an average rainfall in May of 2.64 centimeters with a standard deviation of 0.66 centimeters during the past 10 years. Find a 95% confidence interval for the difference of the true average rainfalls in these two regions, assuming that the observations come from normal populations with different variances. Est : PLBautista

Paired Observations  Used when the samples are not selected independently  The t-value has n-1 degrees of freedom  n represents the no. of pairs Est : PLBautista

Example 8  Twenty college freshmen were divided into 10 pairs, each member of the pair having approximately the same IQ. One of each pair was selected at random and assigned to a statistics section using a transformative method of teaching. The other member of each pair was assigned to a section in which the professor lectured. At the end of the trimester, each group was given the same examination and the following grades were recorded: Est : PLBautista

Example 8 PairTransformativeLecture Find a 98% confidence interval for the true difference in the two learning procedures. Est : PLBautista

Estimation for the Difference of Two Proportions  Best point estimate:  A (1-α)100% confidence interval for p 1 -p 2 is given by Est : PLBautista

Example 9  A poll is taken among the residents of a city and the surrounding county to determine the feasibility of a proposal to construct a civic center. If 2400 of 5000 city residents favor the proposal and 1200 of 2000 county residents favor it, find a 90% confidence interval for the true difference in the fractions favoring the proposal to construct the civic center. Est : PLBautista

Estimation for the Ratio of Two Variances  A (1-α)100% confidence interval for the ratio of two variances is given by Est : PLBautista

Example 10  A standardized placement test in mathematics was given to 25 boys and 16 girls. The boys made an average grade of 82, with a standard deviation of 8, while the girls made an average grade of 78 with a standard deviation of 7. Find a 98% confidence interval for σ 1 2 /σ 2 2, where σ 1 2 and σ 2 2 are the variances of the populations of grades for all boys and girls, respectively, who at some time have taken or will take this exam. Assume the populations to be normally distributed. Est : PLBautista