“Analyzing Computer Use for Leisure Across Different Age Groups: A Comparative Study

QUESTION

The Bureau of Labor Statistics’ American Time Use Survey showed that the amount of time spent using a computer for leisure varied greatly by age. Individuals age 75 and over averaged 0.30 hour (18 minutes) per day using a computer for leisure. Individuals ages 15 to 19 spend 0.8 hour per day using a computer for leisure. If these times follow an exponential distribution, find the proportion of each group that spends: a. Less than 16 minutes per day using a computer for leisure. (Round your answers to 4 decimal places.) b. More than 2 hours. (Round your answers to 4 decimal places.) c. Between 32 minutes and 96 minutes using a computer for leisure. (Round your answers to 4 decimal places.) d. Find the 26th percentile. Seventy-four percent spend more than what amount of time? (Round your answers to 2 decimal places.

ANSWER

“Analyzing Computer Use for Leisure Across Different Age Groups: A Comparative Study”

In the Bureau of Labor Statistics’ American Time Use Survey, the amount of time spent using a computer for leisure was found to vary significantly by age, with individuals aged 75 and over spending an average of 0.30 hours (18 minutes) per day on leisure computer use, while those aged 15 to 19 spent 0.8 hours per day.

To find the proportion of each group that spends less than 16 minutes per day using a computer for leisure, we need to calculate the cumulative distribution function (CDF) for the exponential distribution for each age group.

For individuals aged 75 and over:
CDF(x) = 1 – e^(-λx)
Given x = 16 minutes (0.27 hours) and λ (rate parameter) is the reciprocal of the mean, which is 1/0.30 = 3.33, we can calculate:
CDF(0.27) = 1 – e^(-3.33 * 0.27) ≈ 0.0888

For individuals aged 15 to 19:
Given x = 16 minutes (0.27 hours) and λ (rate parameter) is the reciprocal of the mean, which is 1/0.8 = 1.25, we can calculate:
CDF(0.27) = 1 – e^(-1.25 * 0.27) ≈ 0.2714

To find the proportion of each group that spends more than 2 hours (120 minutes) using a computer for leisure, we need to calculate the complementary CDF (1 – CDF) for each age group.

For individuals aged 75 and over:
Given x = 120 minutes (2 hours) and λ (rate parameter) is 3.33, we can calculate:
1 – CDF(2) = 1 – (1 – e^(-3.33 * 2)) ≈ 0.6332

For individuals aged 15 to 19:
Given x = 120 minutes (2 hours) and λ (rate parameter) is 1.25, we can calculate:
1 – CDF(2) = 1 – (1 – e^(-1.25 * 2)) ≈ 0.4161

To find the proportion of each group that spends between 32 minutes (0.53 hours) and 96 minutes (1.6 hours) using a computer for leisure, we can subtract the CDF at 96 minutes from the CDF at 32 minutes for each age group.

For individuals aged 75 and over:
CDF(1.6) = 1 – e^(-3.33 * 1.6) ≈ 0.4341
CDF(0.53) = 1 – e^(-3.33 * 0.53) ≈ 0.1848
Proportion = CDF(1.6) – CDF(0.53) ≈ 0.4341 – 0.1848 ≈ 0.2493

For individuals aged 15 to 19:
CDF(1.6) = 1 – e^(-1.25 * 1.6) ≈ 0.3509
CDF(0.53) = 1 – e^(-1.25 * 0.53) ≈ 0.1353
Proportion = CDF(1.6) – CDF(0.53) ≈ 0.3509 – 0.1353 ≈ 0.2156

To find the 26th percentile, we need to find the value (x) for which the CDF is equal to 0.26.

For individuals aged 75 and over:
0.26 = 1 – e^(-3.33 * x)
e^(-3.33 * x) = 0.74
-3.33 * x = ln(0.74)
x ≈ ln(0.74) / -3.33 ≈ 0.309

Seventy-four percent spend more than what amount of time?

For individuals aged 75 and over:
74% = 0.74
x = -ln(1 – 0.74) / 3.33 ≈ 0.275 hours (16.5 minutes)

In conclusion, the proportion of time spent using a computer for leisure varies among different age groups, and these calculations provide valuable insights into the patterns of computer use across age demographics. The exponential distribution allows us to make informed estimates and comparisons regarding leisure computer use.