Basic statistics
You already know the mean, the median and the standard deviation from 4th ESO. In Bachillerato tecnológico you will use them constantly as building blocks of larger formulas (covariance, correlation, regression), so it is worth pinning them down rigorously: which properties they satisfy, why they satisfy them, and how data from two variables are combined in a single table.
Mean, variance and standard deviation: a formal review
Two independent epidemiological studies on body mass index (BMI) were carried out in a region. The first study, with a sample of $40$ people, obtained a mean of $22$ and…
Two independent epidemiological studies on body mass index (BMI) were carried out in a region. The first study, with a sample of $40$ people, obtained a mean of $22$ and a variance of $4$. The second study, with a sample of $60$ people, obtained a mean of $25$ and a variance of $5$. If both studies are combined into a single pooled sample of $100$ people, calculate the mean and variance of this combined sample.
A quality control analyst observes manufacturing errors in 10 production batches. The mean number of errors is $6,5$ and the variance is $2,25$. Later, during an audit,…
A quality control analyst observes manufacturing errors in 10 production batches. The mean number of errors is $6,5$ and the variance is $2,25$. Later, during an audit, they notice that a typographical error recorded $2$ errors instead of $8$ for one of the batches. Calculate the corrected new mean and variance without needing to know the individual data for the other 9 batches.
Two-way tables and marginal distributions
In a sociological study on reading habits, 50 people were surveyed about their sex ($X$: $1$ for Male, $2$ for Female) and whether they read more than one book a month…
In a sociological study on reading habits, 50 people were surveyed about their sex ($X$: $1$ for Male, $2$ for Female) and whether they read more than one book a month ($Y$: $1$ for Yes, $2$ for No). The joint absolute frequency table $n_{ij}$ is as follows: $n_{11} = 10$, $n_{12} = 15$, $n_{21} = 20$, $n_{22} = 5$. Construct the marginal distribution table for the variable $Y$ and calculate its relative frequencies.
You can do it on paper and send a photo with your answer.
A veterinary clinic classifies the dogs seen in a month according to their size ($X$: $1$ = Small, $2$ = Medium, $3$ = Large) and the number of visits to the vet in the…
A veterinary clinic classifies the dogs seen in a month according to their size ($X$: $1$ = Small, $2$ = Medium, $3$ = Large) and the number of visits to the vet in the last year ($Y$: $1, 2, 3$). The joint frequency table $n_{ij}$ is:
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For $X=1$: $n_{11}=12$, $n_{12}=8$, $n_{13}=5$
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For $X=2$: $n_{21}=10$, $n_{22}=15$, $n_{23}=10$
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For $X=3$: $n_{31}=4$, $n_{32}=6$, $n_{33}=10$
Calculate the arithmetic mean and the variance of the marginal distribution of the number of visits ($Y$).
In an agronomy study, the amount of fertiliser applied in kg/ha ($X$) and the crop yield in tonnes/ha ($Y$) are analysed. The following two-way table for the joint…
In an agronomy study, the amount of fertiliser applied in kg/ha ($X$) and the crop yield in tonnes/ha ($Y$) are analysed. The following two-way table for the joint frequencies $n_{ij}$ is available:
| $X$ \ $Y$ | $2$ | $4$ | $6$ |
|---|---|---|---|
| 10 | $1$ | $2$ | $0$ |
| 20 | $0$ | $3$ | $1$ |
| 30 | $0$ | $1$ | $2$ |
Calculate the mean and standard deviation of the crop yield conditional on $20$ kg/ha of fertiliser having been applied (that is, from the distribution $Y | X=20$).
Statistical independence
In a study on the relationship between the number of weekly hours spent playing sports ($X$) and the level of work stress ($Y$), the following joint absolute frequency…
In a study on the relationship between the number of weekly hours spent playing sports ($X$) and the level of work stress ($Y$), the following joint absolute frequency table is obtained, where $k$ is an unknown value:
| $X$ \ $Y$ | $2$ | $5$ | $8$ |
|---|---|---|---|
| 0 | $1$ | $2$ | $3$ |
| 3 | $2$ | $4$ | $k$ |
| 6 | $1$ | $2$ | $3$ |
Mathematically determine the value of $k$ so that the variables $X$ and $Y$ are strictly statistically independent.
Two qualitative variables are analysed in a sample of $200$ tourists: continent of origin ($X$: European, American, Asian) and type of accommodation ($Y$: Hotel,…
Two qualitative variables are analysed in a sample of $200$ tourists: continent of origin ($X$: European, American, Asian) and type of accommodation ($Y$: Hotel, Apartment). The marginal relative frequencies are: European $50\%$, American $30\%$, Asian $20\%$. For accommodation: Hotel $60\%$, Apartment $40\%$. Assuming that both variables are statistically independent, how many tourists in the sample are expected to be American and stay in an apartment at the same time?
Position and shape: quartiles, mode and grouped data
The scores obtained by the 9 members of a rhythmic gymnastics team in a flexibility test are: $16, 17, 16, 18, 16, 17, 19, 16, 17$. Determine the mode, the median and…
The scores obtained by the 9 members of a rhythmic gymnastics team in a flexibility test are: $16, 17, 16, 18, 16, 17, 19, 16, 17$. Determine the mode, the median and the lower quartile ($Q_1$) of this dataset.
At a weather station, the maximum daily wind speed ($X$, in km/h) has been grouped into the following intervals: $[0, 10)$, $[10, 20)$, $[20, 30)$, $[30, 40)$, with…
At a weather station, the maximum daily wind speed ($X$, in km/h) has been grouped into the following intervals: $[0, 10)$, $[10, 20)$, $[20, 30)$, $[30, 40)$, with absolute frequencies $5, 12, 8, 5$ respectively. Using the class midpoints of each interval, calculate the arithmetic mean and variance of the wind speed.