Statistics: Time Series Analysis: Trends, Seasonality, and Cycles Worksheet

Question 1

A store records its monthly sales for one year. Sales increase from January through December, with small ups and downs from month to month. Does this time series show a trend, seasonality, a cycle, or random variation? Explain your answer.

Accepted Answer

This time series shows an upward trend because the overall pattern increases over time, even though there are small random ups and downs.

Question 2

The table shows ice cream sales for four months: May: 120 cones, June: 180 cones, July: 260 cones, August: 240 cones. Describe the short-term pattern in the data.

Accepted Answer

The data show sales rising from May to July and then decreasing slightly in August. This pattern may suggest a seasonal increase during warmer summer months.

Question 3

A website has more visitors every Friday and Saturday than on other days of the week. This pattern repeats each week. What type of time series pattern is this? Explain.

Accepted Answer

This is seasonality because the pattern repeats at a regular and predictable interval, which is every week.

Question 4

A local economy goes through several years of growth, then several years of decline, and then growth again. The timing is not exactly the same each time. Is this seasonality or a cycle? Explain.

Accepted Answer

This is a cycle because the pattern rises and falls over a long period, but the timing is not fixed or perfectly regular.

Question 5

A company's quarterly revenue is shown: Year 1 Q1: 50, Q2: 60, Q3: 70, Q4: 110; Year 2 Q1: 55, Q2: 65, Q3: 75, Q4: 120; Year 3 Q1: 60, Q2: 70, Q3: 85, Q4: 130. Identify one trend and one seasonal pattern.

Accepted Answer

The trend is upward because revenue generally increases from Year 1 to Year 3. The seasonal pattern is that Q4 is the highest quarter each year.

Question 6

A time series has the values 10, 13, 15, 18, 20, 22, 25, 27. Find the approximate average change per time period from the first value to the last value.

Accepted Answer

The average change per time period is about 2.43 units because the total change is 27 minus 10, which is 17, and there are 7 time intervals. The calculation is 17 divided by 7, which is about 2.43.

Question 7

A school tracks daily absences for 30 school days. Most days have 20 to 25 absences, but one day has 70 absences during a flu outbreak. What term describes the value of 70, and how could it affect analysis?

Accepted Answer

The value of 70 is an outlier or unusual value. It could make the average number of absences look higher than a typical day and may need to be explained separately.

Question 8

The table shows monthly electricity use for a household: January: 900 kWh, February: 850 kWh, March: 700 kWh, April: 600 kWh, May: 650 kWh, June: 800 kWh, July: 1000 kWh, August: 1050 kWh, September: 850 kWh, October: 650 kWh, November: 700 kWh, December: 880 kWh. Describe the seasonal pattern.

Accepted Answer

Electricity use is highest in winter and summer and lower in spring and fall. This suggests seasonal use related to heating in cold months and cooling in hot months.

Question 9

A moving average is often used in time series analysis. Explain why a 3-point moving average can help when data have random short-term fluctuations.

Accepted Answer

A 3-point moving average helps smooth random short-term fluctuations by averaging nearby values. This makes the overall trend easier to see.

Question 10

The number of bicycles sold at a shop is recorded each month for five years. Sales rise every spring and summer and fall every winter. Sales are also slowly increasing from year to year. What two patterns are present?

Accepted Answer

The data show seasonality because sales rise and fall during the same parts of each year. The data also show an upward trend because sales are slowly increasing from year to year.

Question 11

A stock price rises and falls many times over several months with no clear repeating pattern and no clear overall increase or decrease. Which pattern best describes this time series?

Accepted Answer

This time series is best described as random variation because it has ups and downs without a clear trend, fixed seasonality, or long-term cycle.

Question 12

A city wants to forecast water use for next July. The data show that July water use has increased each year for the past 6 years, and July is always one of the highest-use months. What should the city consider when making its forecast?

Accepted Answer

The city should consider both the upward trend over the past 6 years and the seasonal pattern that July is usually a high-use month. A good forecast would likely be higher than recent non-summer months and higher than earlier July values.

Statistics: Time Series Analysis: Trends, Seasonality, and Cycles

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