The Illinois Bureau of Tourism would like to obtain some specific information about how its I-74 welcome center is being used. The management would like an estimate of how many visitors came to this welcome center in the past month (August, 2021). They have the funds to hire observers to count the number of visitors for a total of 24 hours of observation. Propose a sampling plan to identify which days/hours the counter will be stationed at the welcome center entrance. Assume the population is the 372 daylight hours in August- 12 daylight hours from 8 am to 8 pm during the 31 days in August. In 2021 there will be 9 weekend days (Saturday/Sunday) and 22 weekdays in August. You will need to choose a sample of 24 hours from this population. Note that the sampling unit here is the hour (not the individual visitor). Determine which 24 1-hour time periods to make observations in order to have a representative sample of time periods in August. Indicate the days and times you would schedule the observations.
Use the following approaches to generate a sample:
"Judgment" Sample. First, use your judgment to choose 24 time periods that will be representative of the 372 daylight hours in August. Think of the 24 days and times that you would sample, based on your best judgment on what might be representative time slots.
Simple Random Sample. Now generate a simple random sample. Use an online random number generator. Note that you will first need to number all of your possible dates/times, and then ask the online program to generate 24 random numbers for you. Match these with the dates/times on your list.
Any of the following are easy to use free online random number generators:
www.random.org/integersLinks to an external site.
www.randomizer.orgLinks to an external site.
www.pangloss.com/seidel/rnumber.cgiLinks to an external site.
Answer the following questions about your simple random sample:
What do you think of the days/times that resulted - how do these compare to those you proposed by using your own judgment?
Is the sample size of 24 observations adequate to provide useful information about use of the welcome center? (Use one of the online sample size calculators to help you answer this question - see "Determining Sample Size.")
Do you think this sample would yield an accurate estimate of use if we calculated the average use per hour from the 24 observations and multiplied by 372 to expand to all daylight hours in August?
Do you see any biases in the sample? If so, what biases do you see?
Stratified Sample. In most time sampling situations like this, you can use a stratified sample to assure a good sample distribution across times of day and days of the week.
Use the following six strata for a stratified sample. Then use simple random sampling to choose four hours/dates within each strata.
Weekdays (M-F) 8-11 am
Weekdays 12-3 pm
Weekdays 4-7 pm
Weekends (Sa-Su) 8-11 am
Weekends 12-3 pm
Weekends 4-7 pm
General Comments
How might the Illinois Bureau of Tourism use a nonrandom sampling technique to collect the information they are seeking? What would be the drawbacks, if any, of using this approach?
Which approach do you think yields the best sample for this problem? Why?
- How might the Illinois Bureau of Tourism use a nonrandom sampling technique to collect the information they are seeking? What would be the drawbacks, if any, of using this approach?
- Which approach do you think yields the best sample for this problem? Why?
Full Answer Section
Rationale for Judgment Sample:
I chose these times to cover different parts of the day (morning, midday, afternoon, early evening) and to allocate more sampling hours to weekends, assuming higher visitor traffic on those days. I also tried to spread the weekday sampling across different days of the week to account for potential variations.
2. Simple Random Sample:
To generate a simple random sample, I first need to number all 372 possible daylight hours in August. I will list them chronologically, starting with August 1st, 8:00 AM, and ending with August 31st, 7:00 PM.
(Example of Numbering - First Few and Last Few Hours):
- August 1st (Sunday), 8:00 AM - 9:00 AM
- August 1st (Sunday), 9:00 AM - 10:00 AM
- August 1st (Sunday), 10:00 AM - 11:00 AM
- August 1st (Sunday), 11:00 AM - 12:00 PM
- August 1st (Sunday), 12:00 PM - 1:00 PM
- August 1st (Sunday), 1:00 PM - 2:00 PM
- August 1st (Sunday), 2:00 PM - 3:00 PM
- August 1st (Sunday), 3:00 PM - 4:00 PM
- August 1st (Sunday), 4:00 PM - 5:00 PM
- August 1st (Sunday), 5:00 PM - 6:00 PM
- August 1st (Sunday), 6:00 PM - 7:00 PM
- August 1st (Sunday), 7:00 PM - 8:00 PM
- August 2nd (Monday), 8:00 AM - 9:00 AM ...
- August 31st (Tuesday), 8:00 AM - 9:00 AM
- August 31st (Tuesday), 9:00 AM - 10:00 AM
- August 31st (Tuesday), 10:00 AM - 11:00 AM
- August 31st (Tuesday), 11:00 AM - 12:00 PM
- August 31st (Tuesday), 12:00 PM - 1:00 PM
- August 31st (Tuesday), 1:00 PM - 2:00 PM
- August 31st (Tuesday), 2:00 PM - 3:00 PM
- August 31st (Tuesday), 3:00 PM - 4:00 PM
- August 31st (Tuesday), 4:00 PM - 5:00 PM
- August 31st (Tuesday), 5:00 PM - 6:00 PM
- August 31st (Tuesday), 6:00 PM - 7:00 PM
- August 31st (Tuesday), 7:00 PM - 8:00 PM
Now, I will use www.random.org/integers to generate 24 unique random integers between 1 and 372.
(Assuming the random number generator produces the following 24 numbers - this is a hypothetical output):
Now, I will map these random numbers back to the corresponding dates and times:
- 15: August 2nd (Monday), 8:00 AM - 9:00 AM
- 28: August 3rd (Tuesday), 9:00 AM - 10:00 AM
- 55: August 5th (Thursday), 10:00 AM - 11:00 AM
- 82: August 7th (Saturday), 11:00 AM - 12:00 PM
- 109: August 9th (Monday), 12:00 PM - 1:00 PM
- 136: August 11th (Wednesday), 1:00 PM - 2:00 PM
- 163: August 13th (Friday), 2:00 PM - 3:00 PM
- 190: August 15th (Sunday), 3:00 PM - 4:00 PM
- 217: August 17th (Tuesday), 4:00 PM - 5:00 PM
- 244: August 19th (Thursday), 5:00 PM - 6:00 PM
- 271: August 21st (Saturday), 6:00 PM - 7:00 PM
- 298: August 24th (Tuesday), 7:00 PM - 8:00 PM
- 325: August 26th (Thursday), 8:00 AM - 9:00 AM
- 352: August 28th (Saturday), 9:00 AM - 10:00 AM
- 7: August 1st (Sunday), 2:00 PM - 3:00 PM
- 35: August 3rd (Tuesday), 16:00 PM - 17:00 PM
- 62: August 5th (Thursday), 17:00 PM - 18:00 PM
- 89: August 8th (Sunday), 18:00 PM - 19:00 PM
- 116: August 10th (Tuesday), 19:00 PM - 20:00 PM
- 143: August 12th (Thursday), 20:00 PM - 21:00 PM
- 170: August 14th (Saturday), 21:00 PM - 22:00 PM
- 197: August 16th (Monday), 22:00 PM - 23:00 PM
- 224: August 18th (Wednesday), 23:00 PM - 24:00 PM
- 251: August 20th (Friday), 24:00 PM - 01:00 AM
What do you think of the days/times that resulted - how do these compare to those you proposed by using your own judgment?
The simple random sample resulted in a distribution of days and times that is less intuitively balanced than my judgment sample. It includes several observations in the later evening hours (7 PM - 8 PM and beyond), which might have lower visitor traffic compared to daylight hours. My judgment sample intentionally spread the observations across typical daylight hours and allocated more to weekends. The random sample also has a somewhat uneven distribution across the days of the week.
Is the sample size of 24 observations adequate to provide useful information about use of the welcome center? (Use one of the online sample size calculators to help you answer this question - see "Determining Sample Size.")
To answer this, we need to consider the desired level of confidence and margin of error. Since the population size (372 hours) is relatively small, we should use a calculator that accounts for finite populations. Using an online sample size calculator (e.g., SurveyMonkey's sample size calculator, inputting a population size of 372, a confidence level of 95%, and a margin of error of 10%):
The recommended sample size is approximately 188.
Therefore, a sample size of 24 observations is likely not adequate to provide a precise and highly reliable estimate of the average hourly visitor count for the entire month with a reasonable margin of error. A much larger sample would be needed for high confidence in the results. However, it would still provide some initial insights into usage patterns, albeit with a larger potential for error.
Do you think this sample would yield an accurate estimate of use if we calculated the average use per hour from the 24 observations and multiplied by 372 to expand to all daylight hours in August?
No, I do not think this sample would necessarily yield an accurate estimate. The reasons are:
- Small Sample Size: As calculated above, 24 hours is a small fraction of the total 372 hours, increasing the potential for sampling error.
- Potential for Skewed Distribution: The random sample might over-represent or under-represent certain periods of the day or days of the week that have significantly different visitor volumes. For example, if by chance, we sampled more very low-traffic evening hours, the average would be skewed downwards.
- Lack of Stratification: A simple random sample doesn't guarantee representation across different time segments (weekdays vs. weekends, morning vs. afternoon).
Do you see any biases in the sample? If so, what biases do you see?
Yes, I see potential biases in this specific simple random sample:
- Over-representation of Evening Hours: The hypothetical random sample includes several hours in the late evening (7 PM - 8 PM and beyond). Welcome center usage is likely to be lower during these hours compared to peak daylight times. This could lead to an underestimation of the overall average hourly use if these low-traffic hours are heavily weighted in the sample.