Estimation
Estimation
Estimation
Estimation
We
wish to estimate a characteristic of the population, by using information from the sample Types of Estimates
ESTIMATION OF PARAMETERS
One Population
One
Two Populations
Difference
Mean
Case 1 Case 2
Means
of Two
One
Difference
CASE 1: is known, OR n 30
The
MARGIN OF ERROR
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.
the sample mean is used as an estimate of , we can be (1-)100% confident that the error will not exceed E when the sample size is
How
MARGIN OF ERROR
The
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 midsized cars have a normal distribution, find a 99% confidence interval for the population mean.
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?
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.
MARGIN OF ERROR
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.
we use the sample proportion as an estimate of p, we can be (1 )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?
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.
The
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 46.0. Find a 95% confidence interval for the variance of all such cans assuming that volume is normally distributed.
ESTIMATION OF PARAMETERS
One Population
One
Two Populations
Difference
Mean
Case 1 Case 2
Means
of Two
One
Difference
MARGIN OF ERROR
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.
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.
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.
Paired Observations
Used
The
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:
Example 8
Pair 1
2 3 4 5 6 7 8 9 10
Transformative 76
60 85 58 91 75 82 64 79 88
Lecture 81
52 87 70 86 77 90 63 85 83
Find a 98% confidence interval for the true difference in the two learning procedures.
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 .
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 12/22, where 12 and 22 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.