Exercises: Point estimation: estimators, bias and unbiased sample variance

To estimate the mean height of students in $2.º$ Baccalaureate in a city, a random sample of $5$ students is taken, with the following results in centimetres: $165$, $170$, $172$, $168$, $175$.

• a) Calculate the point estimate of the population mean.

• b) Explain why this value is only an estimate and not the true mean.

In a survey of $300$ randomly chosen students, $87$ state that they cycle to school.

• a) What is the point estimate of the population proportion of students who cycle to school? Calculate it.

• b) Express the result as a percentage and interpret it.

A random sample from a population yielded the values $4$, $6$, $5$ and $9$.

• a) Calculate the sample mean.

• b) Calculate the sample variance (dividing by $n$) and the unbiased sample variance (dividing by $n - 1$).

• c) Which of the two is used as an unbiased estimator of the population variance? What does "unbiased" mean?

True or false, reasoning your answer: "The sample mean is an unbiased estimator of the population mean, meaning that if we were to take many samples of the same size and calculate their means, the average of all of them would equal the population mean."

Seán conducted a small survey of $5$ classmates on the number of hours per week they spend studying Mathematics, obtaining the values: $4$, $6$, $5$, $8$, and $7$. To estimate the dispersion of all students at his school, he calculates the sample mean $\bar{x} = 6$ hours and then computes $s^2 = \frac{(4-6)^2 + (6-6)^2 + (5-6)^2 + (8-6)^2 + (7-6)^2}{5} = \frac{4 + 0 + 1 + 4 + 1}{5} = 2$. In his work, Seán concludes: "The unbiased estimator of the population variance of study hours is exactly $2$ square hours".

1. a) Identify the conceptual error Seán makes in considering $s^2$ an unbiased estimator of the population variance $\sigma^2$.
2. b) Explain why the sample variance calculated by Seán tends to yield a value that underestimates the actual population variance. What role do deviations from the sample mean play in this underestimation compared to deviations from the population mean $\mu$?
3. c) Correct Seán's calculation by finding the value of the corresponding unbiased estimator (the sample variance with Bessel's correction $\hat{s}^2$) and calculate the sample standard deviation with Bessel's correction $\hat{s}$.

Met Éireann has recorded the following maximum temperatures (in ºC) during $6$ consecutive winter days in a town in the Irish midlands: $8$, $12$, $7$, $11$, $9$, and $7$. We want to use this sample to make a point estimate of the population mean and thermal variability for that period.

1. a) Before performing any formal calculations, make a reasoned preliminary estimate: indicate a narrow interval where you believe with certainty that the sample mean $\bar{x}$ will fall, and argue whether you expect the sample standard deviation $\hat{s}$ to be closer to $1$ ºC, $2$ ºC, or $5$ ºC, looking solely at the range (difference between the maximum and minimum values) of the data.
2. b) Calculate the unbiased point estimate of the population mean temperature ($\bar{x}$) and the unbiased point estimate of the population variance ($\hat{s}^2$, sample variance). Round the results to two decimal places.
3. c) Check whether the obtained values match your initial forecast from part a). From the calculated sample variance, find the sample standard deviation $\hat{s}$ and explain why, despite correcting the bias in the variance, $\hat{s}$ is generally not a strictly unbiased estimator of $\sigma$.

The full charging times (in minutes) of $8$ mobile phones of the same brand have been measured: $74$, $81$, $69$, $78$, $85$, $72$, $77$, $80$.

• a) Estimate the mean charging time for this model using an unbiased estimator.

• b) Estimate its variance using an unbiased estimator (unbiased sample variance).

• c) Also calculate the unbiased sample standard deviation and interpret both results.

In a random sample of $500$ customers of a supermarket chain, $210$ state that they use the chain's application to do their shopping.

• a) Estimate the population proportion of customers who use the application.

• b) Calculate the maximum error of the estimate with a confidence level of $95\ \%$, using $E = 1.96 \times \sqrt{\frac{\hat{p} \times (1 - \hat{p})}{n}}$.

• c) Interpret the result: between which values does the true proportion lie, with that level of confidence?

A company wants to estimate the mean of a population and an analyst proposes taking the maximum value of the sample as an "estimator".

• a) Consider the population consisting of the values $\{2, 4, 6\}$ and all samples of size $2$ with replacement. Calculate the population mean and the average of the maximums of all possible samples.

• b) In light of the result, is that estimator unbiased? Explain what the bias of an estimator is based on your calculations.

Consider the population consisting of the five values $\{1, 2, 3, 4, 5\}$.

• a) Calculate the population mean $\mu$.

• b) Write down all samples of size $2$ without replacement and without regard to order, and calculate the mean of each one.

• c) Calculate the average of these sample means and check that it equals $\mu$. Which property of the "sample mean" estimator does this result illustrate?

In a study on the cost of the daily lunch special in an office district of Cork, a student named Cian collects the prices from a random sample of $n$ restaurants. After entering the data into a spreadsheet, he finds that the sum of the squared differences from the sample mean is $\sum_{i=1}^n (x_i - \bar{x})^2 = 36$. However, a computer glitch deletes the list of original data and only leaves recorded that the sample variance $\hat{s}^2$ exceeds the unadjusted sample variance $s^2$ by exactly $0.5$ €$^2$.

1. a) Write the equation that relates the unadjusted sample variance $s^2$, the sample variance $\hat{s}^2$, and the sample size $n$ from the available sum of squares, setting their difference equal to $0.5$.
2. b) Solve the equation to reconstruct the original sample size ($n$) and determine the exact numerical values of $s^2$ and $\hat{s}^2$.
3. c) If the sample size had been ten times larger while keeping the average dispersion per data point identical, what would happen to the difference $\hat{s}^2 - s^2$? Reason what practical implication this has when working with large samples in sociology or economics.

According to data from the Central Statistics Office (CSO), the monthly number of intercity trips made by the young working population is being investigated. A first pilot sample of $4$ young people is drawn, yielding the following data: $1$, $3$, $5$, and $7$ trips. Later, to check the stability of the estimators, $4$ additional observations that present identical relative dispersion are collected, forming an expanded sample of $8$ young people: $1$, $3$, $5$, $7$, $1$, $3$, $5$, and $7$.

1. a) For each of the two samples ($n=4$ and $n=8$), calculate: the sample mean ($\bar{x}$), the unadjusted sample variance ($s^2$), and the sample variance ($\hat{s}^2$).
2. b) Model the bias of the unadjusted sample variance mathematically by analyzing the ratio $\frac{\hat{s}^2}{s^2}$ in both cases. In which of the two samples is the bias of $s^2$ more pronounced? What percentage of undervaluation does $s^2$ represent compared to $\hat{s}^2$ when $n=4$ and when $n=8$?
3. c) Interpret the result in the context of official survey design: why can statistical agencies afford to use uncorrected variance formulas when working with thousands of respondents, but must be very rigorous with the sample variance in small-group studies?

To estimate the mean $\mu$ of a normal distribution, three estimators based on a random sample are proposed: the sample mean, the sample median and the midpoint between the minimum and the maximum of the sample.

• a) Explain what it means for an estimator to be unbiased and discuss whether all three are expected to be unbiased in a symmetric distribution such as the normal.

• b) Explain what the efficiency of an estimator is and reason which of the three will, in general, be less efficient. Why is the sample mean usually preferred?

A property portal takes a sample of $6$ advertisements for rooms to rent in a neighbourhood, with the following monthly prices in euros: $450$, $520$, $480$, $510$, $465$, $495$.

• a) Estimate the mean price of the neighbourhood using an unbiased estimator.

• b) Estimate the variability of the prices by calculating the unbiased sample variance and the unbiased sample standard deviation.

• c) Explain, using the numbers from this exercise, the difference between dividing by $n$ and by $n - 1$, and why the unbiased estimation uses $n - 1$.

From an urn containing an unknown proportion $p$ of red balls, $40$ balls are drawn with replacement, and $14$ are red.

• a) Give the point estimator of $p$ and its value.

• b) Reason why, if instead of $40$ draws $400$ were made, the standard error of the estimate would be divided by approximately $\sqrt{10}$. Use the formula for the error $E = z_{\alpha/2} \times \sqrt{\frac{\hat{p} \times (1 - \hat{p})}{n}}$.

• c) What is the relationship between this property and the law of large numbers?

Let $(x_1, x_2)$ be a simple random sample of size $2$ from a population with mean $\mu$.

• a) Show that the estimator $T = \frac{x_1 + x_2}{2}$ is unbiased for $\mu$, that is, its expected value is $\mu$.

• b) Show that the estimator $S = \frac{x_1 + x_2}{3}$ is biased and calculate its bias in terms of $\mu$.

• c) Propose how you would correct $S$ to make it unbiased.

To estimate the mean waiting time $\mu$ (in minutes) at bus stops in an urban bus network, a simple random sample of size $n=5$ is planned, yielding the independent times $X_1, X_2, X_3, X_4, X_5$, all with the same mean $\mu$ and finite variance $\sigma^2$. The technical committee is debating between three methods (estimators) to estimate $\mu$:

• Method 1: The standard sample mean, $\bar{X} = \frac{1}{5}(X_1 + X_2 + X_3 + X_4 + X_5)$.

• Method 2: A weighted estimator that gives more weight to the intermediate observations, considering that the first and last records suffer from surveyor oversights: $\hat{\mu}_B = \frac{1}{8}(X_1 + 2X_2 + 2X_3 + 2X_4 + X_5)$.

• Method 3: A quick estimator that ignores the intermediate observations and only averages the first and last records: $\hat{\mu}_C = \frac{1}{2}(X_1 + X_5)$.

1. a) Check analytically whether all three methods provide unbiased estimators of the population mean $\mu$.
2. b) Knowing that for independent variables $\text{Var}(aX + bY) = a^2\text{Var}(X) + b^2\text{Var}(Y)$, calculate the variance of each of the three estimators in terms of $\sigma^2$.
3. c) Based on the concept of statistical efficiency, compare the three methods and conclusively justify which one is the best estimator for this study and which one is the worst. What is the cost of choosing Method 3 over Method 1 in terms of estimation precision?

In the presentation of a regional study on the impact of remote working in an Irish county, a junior consultant (Aoife) presents the following three conclusions to the county council:

1. "To estimate the proportion $p$ of remote workers in the county, we surveyed a sample of $n=100$ people and obtained $\hat{p} = 0.22$. Since the CSO census data indicate that actual remote working is $18\,\%$, our estimator $\hat{p}$ has a bias of $+0.04$."

2. "If we quadruple the sample by surveying $n=400$ people, the maximum admissible margin of error $E = z_{\alpha/2}\times\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$ will be divided by $4$, increasing our precision fourfold."

3. "When calculating the dispersion of the weekly hours worked remotely, I used the sample variance $\hat{s}^2$ dividing by $n-1 = 99$. Therefore, by calculating its square root $\hat{s} = \sqrt{\hat{s}^2}$, we have obtained a strictly unbiased estimator of the population standard deviation $\sigma$."

As a statistics specialist reviewing the report:

1. a) Refute conclusion 1 by rigorously distinguishing between the estimation error in a specific sample and the concept of an estimator's bias.
2. b) Correct conclusion 2 by demonstrating analytically, using the margin of error formula, the exact factor by which $E$ is reduced when the sample size increases from $n$ to $4n$.
3. c) Justify why conclusion 3 is false in light of the properties of estimators, explaining why the square root of an unbiased estimator of variance does not, in general, inherit the property of unbiasedness for the standard deviation.
4. d) Write a corrected and technically sound version for each of Aoife's three conclusions.

Unit Challenge. How can socioeconomic reality be estimated without conducting a full census? You are going to investigate the average income of towns in your province (Leinster, Munster, Connacht or Ulster) using open data from the CSO.

Challenge phases:

1. Data collection and sampling: Download a table from the CSO with at least $40$ towns from your province and their average incomes. In a spreadsheet, draw two samples of size $n = 10$: one using simple random sampling and the other stratified (for example, by towns with more or fewer than $10\,000$ inhabitants).

2. Point estimation: In each sample, calculate the sample mean $\bar{x}$, the sample variance, and the sample variance with Bessel's correction $s^2$. Compare the results obtained with the true population parameters ($\mu$ and $\sigma^2$) of the entire set of $40$ towns.

3. Analytical report: Prepare a short report with a comparative table where you theoretically justify why the sample variance with Bessel's correction is an unbiased estimator of $\sigma^2$, and evaluate which of the two sampling designs provided an estimate of $\mu$ with a lower percentage error (for example, below $3.5\,\%$).

Evaluation criteria:

• Rigour and representativeness in the design of both sampling procedures.

• Mathematical accuracy in calculating estimators and sample variance with Bessel's correction using a spreadsheet.

• Conceptual clarity when justifying bias and comparing estimates against the true parameters.