Estimation @ Confidence Intervals for One

Population MeanChapter 8

Estimating a Population Mean

Example (Point Estimate of a Population Mean): The U.S Census Bureau publishes annual price figures for new mobile homes. The figures are obtained from sampling not from census. A simple random sample of 36 new mobile homes yielded the prices, in thousands of dollars, show in the table. Use the data to estimate the population mean price, μ, of all new mobile homes.

63.8 64.4 55.2 52.9 59.9 58.2 51.6 68.9 58.6

63.1 69.4 59.7 53.7 62.7 57.7 51.5 45.3 68.9

45.9 52.5 67.2 55.1 60.3 60.0 51.9 47.3 49.7

52.0 72.7 72.8 56.6 70.5 53.9 66.4 59.8 73.9

Solution:We estimate the population price, μ, of all new mobile homes by the sample mean price, , of the 36 new mobile homes technology.

Based on the sample data, we estimate the mean price, μ, of all new mobile homes to be approximately $59.28 thousand, that is $59,280.An estimate of this kind is called a point estimate for μ because it consists of a single number or point.

Confidence Interval for One Population Mean (σ Is Known)

Confidence Intervals for One Populations Mean (σ Is Known)

• To begin, we introduce some general notation used with confidence intervals. Frequently, we want to write the confidence level in the form 1 - , where is a number between 0 and 1; that is, if the confidence level is expressed as a decimal, is the number that must be subtract the confidence level from 1. If the confidence level is 95.44%, then = 1 – 0.9544 = 0.0456.

• Next, recall from Chapter 6, that the symbol Z denotes the z-score that has area to its right under the standard normal curve. So, for example, Z0.05 denotes the Z-score that has area 0.05 to its right and Z/2 denotes the z- score that has area /2 to its right.

When to Use?

Example: The U.S. Bureau of Labor Statistics collects information on the ages of people in the civilian labor force and publishes the results in Employment and Earnings. Fifty people in the civilian labor force are randomly selected; their ages are displayed in Table 8.3. Find a 95% confidence interval for the mean age, μ, of all people in the civilian labor force. Assume that the population standard deviation of the ages is 12.1 years.

Solution:The sample size is 50, which is large, and the population standard deviation is known, we can use One-Mean-z-Interval Procedure to find the required confidence interval.Step 1: For confidence level of 1-α, use Table II to find Zα/2

95% confidence level, so α =1-0.95 = 0.05From Table II : Zα/2 = Z0.05/2 = Z0.025 = 1.96

Step 2: The confidence interval for μ is from;1–0.025 = 0.975 (Find in Table II)

We know = 12.1, n = 50, Z/2 = 1.96.

Find sample mean :

95% confidence interval for is from:

Step 3: Interpret the confidence levelWe can be 95% confident that the mean age,μ,of all people in the civilian labor force is somewhere between 33.0 years and 39.8 years

Margin of Error for the Estimate of

Example: Based on a sample of 30 randomly selected years, a 90% confidence interval for the mean annual precipitation in one city is from 47.8 inches to 50.2 inches. Find the margin of error.

Solution:

Margin of error, E = (50.0 - 47.8)/2 =1.247.8 50.2

50.2 - 47.8 = 2.4

E= 2.4/2 =1.2 E= 2.4/2 =1.2

Determining the Required Sample Size (n)

Example: Refer last example (The U.S Bureau)Consider again the problem of estimating the mean age, , of all people in civilian labor force.

1.Determine the sample size needed in order to be 95% confident that μ is within 0.5 year of the estimate, . Recall that σ=12.1 years.

1.Find a confidence interval for μ if a sample of the size determined in part (a) has mean age of 38.8 years.

Solution (a)We use formula given. We know that = 12.1 and E = 0.5. The confidence level is 0.95, which means that = 0.05 and z /2 = z0.025 = 1.96. Thus

which, rounded up to the nearest whole number, is 2250.

InterpretationIf 2250 people in the civilian labor force are randomly selected, we can be 95% confident that the mean age of all people in the civilian labor force is within 0.5 year of the mean age of the people in the sample.

Solution (b)• Applying z-interval procedure with = 0.05, • = 12.1, = 38.8, and n = 2250, we get the confidence

interval

InterpretationWe can be 95% confident that the mean age, , of all people in the civilian labor force is somewhere between 38.3 years and 39.3 years.

Confidence Interval for One Population Mean (σ Is Unknown)

• t-distribution

Confidence Interval for One Population Mean (σ Is Unknown)

• t-curve

Example For a t-curve with 13 degrees of freedom (df), determine t

0.05 ; that is, find the t-value having area 0.05 to its right, as shown in below figure:

Use Table IV to find the value of t 0.05

SolutionTo find the t-value in question, we use Table IV,degrees of freedom (df)= 13 and t0.05 ,

So, the value of t0.05 with df=13 is 1.771

Obtaining Confidence Intervals for a Population Mean When is Unknown

Example: A laboratory tested a sample of twelve chicken eggs and found that the mean amount of cholesterol was 186 milligrams and stand deviation 19.0 milligrams. Construct a 95% confidence interval for the true mean cholesterol content of all such eggs. Assume that this data don’t have any outliers.

SolutionPopulation standard deviation, σ is unknown.Only sample standard deviation is known s=19.= 186n= 12 so df = n-1=12-1 = 11

Step 1:For confidence level of 1- α, use Table IV to find tα

α=1 – 0.95 = 0.05 so α/2=0.025 t0.025 with df =11 is 2.201

Solution (continue)Step 2: The confidence interval for μ is from

Step 3: Interpret the confidence intervalWe can be 95% confident that the mean amount of cholesterol is somewhere between 173.9 milligrams and 198.1 milligrams.