Suppose you are conducting a study about how the average US worker spends time over the course of a workday. You are interested in how much time workers spend per day on personal calls, emails, and social networking websites, as well as how much time they spend socializing with coworkers versus actually working. The most recent census provides data for the entire population of US workers on variables such as travel time to work, time spent at work, and break time at work. The census, however, does not include data on the variables you are interested in, so you obtain a random sample of 74 full-time workers in the United States and ask about personal calls, emails, and so forth. You are curious about how your sample compares with the census, so you also ask the workers the same questions about work that are asked in the census. Suppose the mean time spent at work per day from the most recent census is 8.41 hours, with a standard deviation of 0.86 hours. Your sample of 74 US workers provides a mean time spent at work per day of 8.65 hours with a sample standard deviation of 1.05 hours. Organize this information by completing the following table. μ = M = σ = s = σM M = sM M

Let's organize the given information and complete the table using the appropriate formulas.

Provided information:

• Census population mean time spent at work per day (μ) = 8.41 hours
• Sample mean time spent at work per day (M) = 8.65 hours
• Census population standard deviation (σ) = 0.86 hours
• Sample standard deviation (s) = 1.05 hours
• Sample size (n) = 74

Now let's fill out the table:

Symbol	Description	Value
μ	Population mean	8.41 hours
M	Sample mean	8.65 hours
σ	Population standard deviation	0.86 hours
s	Sample standard deviation	1.05 hours
σ<sub>M</sub> = $\frac{\sigma}{\sqrt{n}}$	Standard error of the population mean	$\frac{0.86}{\sqrt{74}}$ ≈ 0.10 hours
s<sub>M</sub> = $\frac{s}{\sqrt{n}}$	Standard error of the sample mean	$\frac{1.05}{\sqrt{74}}$ ≈ 0.12 hours

Explanation:

• Standard error of the population mean (σ_M) is calculated by dividing the population standard deviation by the square root of the sample size:
  $σ_M = \frac{0.86}{\sqrt{74}} ≈ 0.10 \, \text{hours}$

• Standard error of the sample mean (s_M) is calculated by dividing the sample standard deviation by the square root of the sample size:
  $s_M = \frac{1.05}{\sqrt{74}} ≈ 0.12 \, \text{hours}$

Would you like more details on any part of the process or have any questions?

Related Questions:

1. How is the standard error of the mean useful in comparing population and sample data?
2. What assumptions must be met to calculate the standard error in this manner?
3. How could we test if the difference between the population mean and sample mean is statistically significant?
4. What could account for the difference between the census mean and the sample mean?
5. How would increasing the sample size affect the standard error?

Tip:

Standard error decreases as the sample size increases, making the sample mean a more accurate estimate of the population mean.