Lift Lines and Snowfall: Sample Estimates

Use random samples to estimate population values and compare how estimates vary.

1. Estimate Lift-Line Waits

Use the random sample to make estimates. Treat each recorded lift-line wait as one observation.

A ski resort recorded 1,200 lift-line waits on Saturday. Priya randomly selected 40 of those records. The table shows all 40 sampled waits.

Random Sample of Saturday Waits
Wait (minutes)	Number of records
2	4
4	8
6	14
8	10
10	4

What percent of the sampled waits lasted at least 8 minutes? %

Use the sample to estimate how many of Saturday's 1,200 waits lasted at least 8 minutes. waits

Find the mean wait in the sample. minutes

How should the resort use the sample mean when describing all 1,200 recorded waits?

A. Report it as the longest of all recorded waits.B. Report it as the wait for every person in the population.C. Report it as the exact mean of all recorded waits.D. Report it as an estimate of the mean of all recorded waits.

2. Compare Lift-Line Estimates

Each sample below was selected separately at random from the same 1,200 Saturday records. Records may appear in more than one sample.

Four Random Samples
Sample	Records sampled	Waits of 8+ minutes
Pine	40	12
Summit	40	18
Ridge	40	14
Basin	40	16

Estimate the number of waits lasting at least 8 minutes among all 1,200 records using each named sample. Pine: waits; Summit: waits

Use all four samples to make four population-count estimates. What is the difference between the highest and lowest estimates? waits

The resort averages the four sample percentages to create one planning estimate.

Using that average percentage, how many of the 1,200 waits are estimated to last at least 8 minutes? waits

Why can these four random samples produce different estimates?

A. Different records can be selected by chance in each sample.B. The total number of Saturday records changes with each sample.C. Every random sample must contain exactly the same records.D. Different estimates prove that none of the samples was random.

The manager will review staffing if any sample-based estimate exceeds 500 waits lasting at least 8 minutes. Which statement correctly applies this rule?

A. A review is needed because every estimate exceeds 500 waits.B. No review is needed because every estimate is below 500 waits.C. A review is needed because the Summit estimate exceeds 500 waits.D. No review is needed because the average estimate is below 500 waits.

3. Compare Snowfall Samples

Use the historical records described below. A sample-based estimate is not an exact count or a forecast.

Mateo studies the resort's daily snowfall records for all 120 days of last season. He selects four separate random samples of 12 days each.

Samples of Last Season's Snowfall
Sample	Days sampled	Days with 10+ cm
One	12	3
Two	12	5
Three	12	4
Four	12	6

10.

Using Sample One, estimate how many of last season's 120 days had at least 10 cm of snowfall. days

11.

Calculate the percent of days with at least 10 cm of snowfall in each sample. What is the difference between the highest and lowest percentages? percentage points

12.

Mateo also takes four random samples of 24 days each from the same season. Compare the variation in the sample percentages in the two sets. Use the difference between the highest and lowest percentages.

Larger Snowfall Samples
Sample	Days sampled	Days with 10+ cm
North	24	8
South	24	9
East	24	10
West	24	9

A. The 24-day samples have less variation in their percentages.B. Neither set has any variation in its percentages.C. The 12-day samples have less variation in their percentages.D. Both sets have the same variation in their percentages.

13.

The four 24-day samples have an average of 9 days with at least 10 cm of snowfall per sample. Which inference is supported?

A. Exactly 9 days in the entire season had at least 10 cm of snowfall.B. About 45 days in the entire season had at least 10 cm of snowfall.C. The next 120-day season will have exactly 45 days with at least 10 cm of snowfall.D. Every group of 24 days had exactly 9 days with at least 10 cm of snowfall.

14.

The four 12-day samples give population estimates from 30 to 60 days with at least 10 cm of snowfall. Mateo says, “The actual count must be between 30 and 60 because all four samples were random.” Is that guaranteed? Explain your reasoning.