- Foundational Questions
- Calculations in Context
- Sample Size Calculations
- Identify the level of confidence (i.e., 100(1-α)%).
- Identify whether a lower confidence bound (greater than H_{A}), upper confidence bound (less than H_{A}), or confidence interval (not equals H_{A}) should be constructed.
- Find Z^{*} (I suggest including a drawing to illustrate your calculation).
- Calculate the confidence region values (i.e., x̄+Z^{*}SE).
- Specifically interpret the confidence region with a complete sentence.
A reminder to use these steps will NOT be provided on future quizzes, but you should get in the habit of following them. See here for a demonstration of the steps.
Calculations in Context
Radius of Jupiter
Researchers measured the equatorial radius of Jupiter 40 different times. The method used is largely without bias but there is measurement-to-measurement variability. In fact, σ is assumed to be 28 km. The mean of the 40 measurements was 71492 km. From this, construct and interpret a confidence interval assuming that α=0.01.
BOD in Effluent
The managers of a wastewater treatment plant monitored the amount of biological oxygen demand (BOD; lbs/day) in the effluent of the plant each month from January 1991 to October 2000. The managers would need to take corrective actions if the average BOD over this time period was significantly greater than 2200 lbs/day at a 10% rejection level. Previous studies indicated that the standard deviation was 1200 lbs/day. Summary statistics from their sample of days is given below. From this construct a proper confidence region.
n Min. 1st Qu. Median Mean 3rd Qu. Max.
118 630 1600 2240 2504 3193 6023
Medical School Admissions
Admissions representatives at the University of Minnesota medical school were concerned that the average grade point average of applicants in non-science courses had dropped below 3.7. A sample of 40 of the most recent applicants indicated that the mean was 3.60. Information from the Association of American Medical Colleges suggested that the overall standard deviation was 0.35. From this construct a proper confidence region for the mean grade point average assuming that α=0.05.
Banff Snow Depth
Hebblewhite et al. (2000) reported the mean snow pack height (in cm) for Banff (data are below). Their results are shown in Table 1. Use these results to compute a 99% confidence interval for the mean snow pack height assuming that σ=15 cm.
Table 1: Summary statistics for the depth of snow in Banff.
n mean sd min Q1 median Q3 max
15.00 44.61 15.15 29.00 32.76 45.51 46.46 80.39
Creatine Phosphate Concentrations
The concentrations (International Units per liter) of creatine phosphokinase (an enzyme related to muscle and brain functions) in 36 male volunteers was recorded and the results are in Table 2. Construct a proper confidence region for the population mean creatine phosphokinase assuming that the H_{A} is a “greater than”, α=0.05, and σ=40.
Table 2: Summary statistics for the creatine phosphate concentrations in 36 male volunteers.
n mean sd min Q1 median Q3 max
36.00 98.28 40.38 25.00 67.75 94.50 118.25 203.00
Brule River Gage Heights
The maximum gage heights (feet) of the Bois Brule River in Brule, WI from 10-25Feb05 were recorded and summarized in Table 3. Use these results to construct a proper confidence region for the population mean gage height assuming that the H_{A} is a “not equals”, α=0.05, and σ=0.20.
Table 3: Summary statistics for the gage height on the Brule River.
n mean sd min Q1 median Q3 max
16.00 1.73 0.20 1.53 1.57 1.65 1.89 2.11
Population Density in Wisconsin Counties
The population density (number of people per acre of land) for 15 randomly selected Wisconsin counties was recorded and is summarized in Table 4. Construct a proper confidence region for the population mean density assuming that the H_{A} is a “less than”, α=0.10, and σ=125.
Table 4: Summary statistics for the population density of Wisconsin Counties.
n mean sd min Q1 median Q3 max
15.00 92.86 126.36 10.20 23.90 52.10 82.60 429.00
Sample Size Calculations
The next two questions are not confidence region questions, rather they are asking you to compute a sample size given σ, a margin-of-error tolerance, and a level of confidence (which, ultimately, is turned into a Z^{*}). See here or the appropriate section in the reading for the formula and example calculations.
Blood Pressure in Children
An investigator wants to estimate the mean systolic blood pressure in children with congenital heart disease who are between the ages of 3 and 5. The investigator plans on using a 99% confidence interval and desires a margin of error of 5 units. The standard deviation of systolic blood pressure is unknown, but the investigator conducted a literature review and found that the standard deviation of systolic blood pressures in children with other cardiac defects is between 15 and 20.[^1]
- Use this information to construct a “worst-case scenario” required sample size.
- How would the required sample size change for the “best-case scenario”.
Calf Growth
Calf growth early in life should be approximately 1000 g per day. Owners of a large cattle farm want to assure that their calves are growing at approximately this rate because slow growth might suggest a feeding problem and fast growth would suggest a future size problem (i.e., too many big animals in the available space). The owners want to sample enough of their calves so that they can estimate the growth rate to within 50 g per day, with 99% confidence, assuming that variabilty between individual calves is approximately 200 g per day. Calculate their required sample size.
Pebble Size
Geologists measure the longest axis of pebbles to determine “grain” sizes. If the standard deviation of pebble long-axis length for a particular site is known to be 4 mm, how many pebbles must be measured in order to determine the average pebble length within 0.1 mm with 99% confidence?
Internet Usage
An investment group wants to start an Internet Service Provider (ISP) and, for their business plan and model, needs to estimate the average Internet usage of households. How many households must be randomly selected to be 95% sure that the sample mean is within 1 minute of the population mean? Assume that a previous survey of household usage had a standard deviation of 6.95 minutes.
Counting Plants in Plots
Suppose that a plant ecologist is to examine a very large tract of land that has been subdivided into 1400 plots of 10 m^{2} (10 square meters). The researcher wants to determine, with 90% confidence, the mean density of plants per plot for the entire tract of land to within 10 plants per plot. A pilot study indicated that the standard deviation was approximately 50 plants per plot. Determine how many 10 m^{2} plots the researcher should examine to reach her stated goals.