Chapter I - Statistics I - Exercises

This is a set of exercises created by the teaching Faculty for Statistics I, Chapter I - Index Numbers, from the Lisbon Accounting and Business School.

Questions

Question 1

Consider the following sales data, in thousands of euros, for a given company:

Year	2020	2021	2022	2023	2024	2025
Sales	500	650	600	800	750	950

1. Build the index series with fix base in 2020.

Year	2020	2021	2022	2023	2024	2025
\(I^{sales}_{t|2020}\)

2. TRUEFALSE We could say that in year 2022, according to the index computed above, sales increased 20% relative to the base year.

3. Study the evolution of sales, between consecutive years.

We need to compute the link or chain index.

Year	2020	2021	2022	2023	2024	2025
Sales	500	650	600	800	750	950
\(I^{sales}_{chain}\)		130	92.3	133.3	93.8	126.7

From 2020 to 2021 sales grew by 30%, from 2021 to 2022 sales diminished 7.7%, from 2022 to 2023 sales grew 33.3%, from 2023 to 2024 sales dropped 6.2%, and from 2024 to 2025 sales grew 26.7%.

4. What happened to sales during the whole studied period?

Sales grew 90%, between 2020 and 2025.

5. The growth rate between 2022 and 2024 was 25%90%14%none of the above

6. The (rounded) average growth rate in the studied period is 18%9%14%None of the above

Question 2

Consider the following sales data, in thousands of euros, for a given company:

Year	2020	2021	2022	2023	2024	2025
Sales	300	250	\(a\)	240	\(b\)	340

1. TRUEFALSE The average growth rate for sales, between 2020 and 2025 was, approximately, 2.53%.

2. Let \(i_{2022|2020}=67\). Also, assume you know that the growth rate doubled between 2023 and 2024. The values for \(a\) and \(b\) are respectively: 201 and 333.1201 and 300240 and 333.1None of the above

3. TRUEFALSE The average growth rate needed to triple sales between 2025 and 2029 would have to be 31.61%.

4. TRUEFALSE Assuming an average growth rate of 10%, it would be necessary to wait for 6 years for sales to triple.

Question 3

A given company presented their financial statements with the following information:

Year	2020	2021	2022	2023	2024	2025
Growth rate of sales		20%	15%	20%	10%	25%
Price Index (base = 2022)	81.17	89.29	100	115	128.8	148.12

1. Write the average growth rate of (nominal) sales for the whole period (rounded up, for example 11 if percentage, or 0.11 if decimal):

2. Find the annual real growth rate for sales, when this was negative.

Start building the index for sales, base 2022, and remember \(i^v_{t|b}=i^q_{t|b}\times i^p_{t|b}\) to build the real growth rates:

Year	2020	2021	2022	2023	2024	2025
Sales Index (base = 2022)	72.46	86.96	100	120	132	165
Price Index (base = 2022)	81.17	89.29	100	115	128.8	148.12
Real Sales Index (base = 2022)	89.27	97.39	100	104.35	102.48	111.4

Clearly, the real growth rate was negative in 2024, and it was -1.79%.

3. How do you interpret the price index for years 2020 and 2021?

From 2020 to 2022 prices grew 23.2 %, and from 2021 to 2022 prices grew 11.99 %.

Question 4

For some country, water consumption in the period 2017 and 2025 evolved according the simple index, base 2022, shown in the following table:

Year	2017	2018	2019	2020	2021	2022	2023	2024	2025
Sales Index (base = 2022)	75.6	78.5	82.5	88.1	93.8	100	104.8	107.7	107.9

1. TRUEFALSE Water consumption betwee 2019 and 2021 grew 12.5%.

2. TRUEFALSE The chain index of 2019 and 2020 are respectively equal to 105.1 and 106.8.

3. Taking yera 2017 as base year, \(i_{2025|2017}\) would be equal to: 136.32142.7175.6None of the above

4. TRUEFALSE The average growth rate for water consumption in the whole period was 4.54%. Overall, during the whole period, water consumption grew 36.32%.

Question 5

Wayne Industries reported sales using the chain index shown in the following table:

Year	2019	2020	2021	2022	2023	2024	2025
Sales Index	120	125	135	145	150	160	165

1. The average growth rate between 2019 and 2025 was approximately: 50%42.86%46.01%None of the above

2. The growth rate for sales between 2018 and 2025 is approximately: 1062.8%65%42.86None of the above

3. TRUEFALSE The average semi-annual growth rate between 2021 and 2024 is approximately 23.10%.

4. TRUEFALSE The value for the semi-annual growth rate, in real terms, if the IPC (fix base) went from 130 in 2021 to 180 in 2024, is approximately equal to 10%.

Question 6

Consider the following data regarding nominal wages for the workers of Watto’s Junkyard. You also have the inflation rate in Tatooine, for the period 2021 and 2025.

Year	2021	2022	2023	2024	2025
Wage Index (base 2022)	93.5	100	104.3	103.8	109.7
Inflation rate	3.0%	2.4%	2.8%	2.3 %	3.0%

1. TRUEFALSE The nominal wage growth rate, from 2021 to 2025, is 17.33%.
2. TRUEFALSE The average growth rate of inflation between 2021 and 2025 is 2%.
3. Fill the following table (use 2 decimal places, for example 101.01):

Year	2022	2023	2024	2025
Linked-Chain Index for wages

4. The year where the wages had the largest real increase was .

Question 7

Consider the following data with respect to the evolution of the Euro Area GDP (EU27) between 2018 and 2024, constant prices (base 2015), as well as the chain index for prices for the same period:

indicator	2018	2019	2020	2021	2022	2023	2024	2025
GDP	11333533	11546104.8	10903404.2	11596747.5	12010512.8	12063464.0	12210051.7	12382215.9
CPI_link	NA	101.4	100.7	102.9	109.2	106.4	102.6	102.5

1. TRUEFALSE The inflation rate between 2022 and 2023 was -2.56%.

2. TRUEFALSE Prices grew from 2021 to 2025 -0.39%.

3. TRUEFALSE The average growth rate between 2025 and 2018, for GDP, was 1.27 % every year.

4. TRUEFALSE In 2024 and 2025 the real and nominal growth rates (for GDP) were the same.

Question 8

The table below has data on regular consumption goods. You will find prices and quantities, before and after austerity measures were put in place for a economically struggling country. In this period, the LQI and PQI were 91.4 and 91.3 respectively.

Products	Before austerity		After austerity
	Price	Quantity	Price	Quantity
Bread (units)	0.2	420	0.25	360
Milk (Lts)	0.7	360	0.80	336

1. Assume the time window, between the two periods, was 3 years. The average growth rate for the price of bread was: %. (use two decimals, for example 9.81)

2. TRUEFALSE According to the Fisher index, in the studied period, the variation of consumed quantities, for both products, was negative and equal to -8.65%.

3. TRUEFALSE The real and nominal growth rates, for sales, for both products, were similar after the implementation of the austerity measures.

4. Suppose that after the implementation of the austerity measures, and until now, the price of milk dropped 5%. Then, the current price for milk would be (again, use two decimal places, for example 0.56)

Question 9

Public Transportation	Year 1		Year 2
	Trips	Price per trip	Trips	Price Index (Base: year 1)
Bus	1,763,521	1.15	1,875,345	108.70
Metro	3,148,350	1.2	3,396,138	112.50

1. TRUEFALSE According to the LPI for year 2, prices for public transportation grew 11.17% compared to year 1.

2. TRUEFALSE The real change, according to PQI in year 2, was 7.35% compared to year 1.

3. The composite index for value, in the public transportation sector, for year 2 compared to year 1, represents what type of change? realnominalquantityprices

Question 10

After tracking a basket with 30 goods for 3 years (2023 to 2025), the following data was collected:

	2023	2024	2025
2023	\(\sum p^k_{2023}q^k_{2023} = 8,562\)	\(\sum p^k_{2023}q^k_{2024}=9,210\)	\(\sum p^k_{2023} q^k_{2025} = 9,920\)
2024	\(\sum p^k_{2024}q^k_{2023} = 9,150\)	\(\sum p^k_{2024}q^k_{2024} = 9,584\)
2025	\(\sum p^k_{2025}q^k_{2023}=9,657\)		\(\sum p^k_{2025}q^k_{2025} = 10,013\)

Fill the following table (use index in hundreds, rounded to the closest integer, for example 107):

	2023	2024	2025
\(LPI_{t|2023}\)	100
\(PPI_{t|2023}\)	100
\(FPI_{t|2023}\)	100

Question 11

An Index Number is used:

To measure the change in prices To measure the change in consumed quantities To measure the change of a variable/phenomena over time To measure the changes in the demand

An index number is expressed as

Ratios Percentages Either of the above None of the above

A price index number, computed with a constant reference is:

A price index A price index (fixed base) Both of the above None of the above

The most adequate average to compute the average growth rate, for example, for prices, is:

Median Arithmetic mean Harmonic mean Geometric mean

The Consumer Price Index (CPI) is computed using:

The Laspeyres formula The Paasche formula Average of Laspeyres and Paasche The Fisher formula

Which of the following passes the test of time reversion?

\[IP_{t|0}=\frac{\sum p_tq_0}{\sum p_0 q_0}\] \[IP_{t|0}=\frac{\sum p_t q_t}{\sum p_0 q_t}\] \[IP_{t|0}=\sqrt{\frac{p_t q_0}{p_0q_0}\times\frac{\sum p_tq_t}{p_0q_t}}\] None of the above

An index number, computed using Fisher’s formula is considered ideal because passes:

The factor reversion test Time reversion test Both test above Circularity test

From the following identities, select the FALSE one

\(IV_{t|0}=LPI_{t|0}\times PQI_{t|0}\) \(IV_{t|0}=CPI_{t|0}\times PQI_{t|0}\) \(IV_{t|0}=FPI_{t|0}\times FQI_{t|0}\) \(IV_{t|0}=PPI_{t|0}\times PQI_{t|0}\)

TRUEFALSE The real growth rate, within a given timeframe, corresponds to the nominal growth rate, deflated by the CPI (consumer price index).