Point estimation: estimators, bias, and sample variance
Nobody can ask every high school student in Mexico how much they spend each month on phone top-ups and data, but you can ask a sample of $100$. Does that sample number "count" as an estimate of the real spending? What formula should be used so that it isn't systematically off? That is what point estimation is about.
What a point estimator is
To estimate the mean height of students in $2.º$ high school in a city, a random sample of $5$ students is taken, with the following results in centimeters: $165$,…
To estimate the mean height of students in $2.º$ high school in a city, a random sample of $5$ students is taken, with the following results in centimeters: $165$, $170$, $172$, $168$, $175$.
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a) Calculate the point estimate of the population mean.
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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…
In a survey of $300$ randomly chosen students, $87$ state that they cycle to school.
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a) What is the point estimate of the population proportion of students who cycle to school? Calculate it.
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b) Express the result as a percentage and interpret it.
Unbiased estimator
The same parameter can be estimated in several ways: with the mean, with the median, with the midpoint between the minimum and the maximum… Not all of them are equally good.
Samples without replacement
If the elements are not returned to the population, each sample of size $2$ is a different pair. If, in addition, order doesn't matter, $(a, b)$ and $(b, a)$ are the same sample and are counted only once: these are the combinations of $n$ elements taken $2$ at a time.
How to prove that an estimator is unbiased
If $x_1, x_2$ are the values of a simple random sample from a population with mean $\mu$, each one has expected value $\mu$ (the "long-run average"). The expected value of a sum is the sum of the expected values, and constants can be factored out.
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…
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."
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…
A company wants to estimate the mean of a population and an analyst proposes taking the maximum value of the sample as an "estimator".
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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.
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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…
Consider the population consisting of the five values $\{1, 2, 3, 4, 5\}$.
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a) Calculate the population mean $\mu$.
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b) Write down all samples of size $2$ without replacement and without regard to order, and calculate the mean of each one.
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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?
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 $\mu$,…
Let $(x_1, x_2)$ be a simple random sample of size $2$ from a population with mean $\mu$.
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a) Show that the estimator $T = \frac{x_1 + x_2}{2}$ is unbiased for $\mu$, that is, its expected value is $\mu$.
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b) Show that the estimator $S = \frac{x_1 + x_2}{3}$ is biased and calculate its bias in terms of $\mu$.
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c) Propose how you would correct $S$ to make it unbiased.
Sample variance: why divide by $n-1$
When estimating the variability of a population from a sample, there is a surprise: dividing by $n$, as in the variance of a data set, gives a biased estimator.
The names vary from one text to another (some materials call $s^2$ the quasi-variance), but what doesn't change is the divisor. Always check what you are dividing by and whether you are being asked to describe the data or to estimate the population.
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…
A random sample from a population yielded the values $4$, $6$, $5$ and $9$.
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a) Calculate the sample mean.
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b) Calculate the sample variance (dividing by $n$) and the unbiased sample variance (dividing by $n - 1$).
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c) Which of the two is used as an unbiased estimator of the population variance? What does "unbiased" mean?
The full charging times (in minutes) of $8$ cell phones of the same brand have been measured: $74$, $81$, $69$, $78$, $85$, $72$, $77$, $80$. a) Estimate the mean…
The full charging times (in minutes) of $8$ cell phones of the same brand have been measured: $74$, $81$, $69$, $78$, $85$, $72$, $77$, $80$.
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a) Estimate the mean charging time for this model using an unbiased estimator.
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b) Estimate its variance using an unbiased estimator (unbiased sample variance).
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c) Also calculate the unbiased sample standard deviation and interpret both results.
A real estate website takes a sample of $6$ advertisements for rooms for rent in a neighborhood, with the following monthly prices in pesos: $9\,000$, $10\,400$,…
A real estate website takes a sample of $6$ advertisements for rooms for rent in a neighborhood, with the following monthly prices in pesos: $9\,000$, $10\,400$, $9\,600$, $10\,200$, $9\,300$, $9\,900$.
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a) Estimate the mean price of the neighborhood using an unbiased estimator.
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b) Estimate the variability of the prices by calculating the unbiased sample variance and the unbiased sample standard deviation.
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c) Explain, using the numbers from this exercise, the difference between dividing by $n$ and by $n - 1$, and why the unbiased estimation uses $n - 1$.
Estimation error and efficiency
A point estimate doesn't say how far it might be from the parameter. For a proportion, the approximate margin of error at a given confidence level is:
Efficiency and the law of large numbers
Among several unbiased estimators, the best one is the one that varies less from one sample to another: the most efficient. In a normal population, the mean, the median, and the midpoint between the minimum and the maximum are unbiased by symmetry, but the midpoint depends only on two extreme values and is the one that fluctuates the most; the mean uses all the data and is the most efficient, which is why it is the standard estimator of $\mu$.
The fact that the error decreases as $n$ grows is exactly what the law of large numbers says: as the sample grows, $\bar{x}$ and $\hat{p}$ get closer and closer to the real value.
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…
In a random sample of $500$ customers of a supermarket chain, $210$ state that they use the chain's application to do their shopping.
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a) Estimate the population proportion of customers who use the application.
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b) Calculate the maximum error of the estimate with a confidence level of $95\ \%$, using $E = 1.96 \times \sqrt{\frac{\hat{p} \times (1 - \hat{p})}{n}}$.
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c) Interpret the result: between which values does the true proportion lie, with that level of confidence?
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…
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.
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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.
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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?
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.…
From an urn containing an unknown proportion $p$ of red balls, $40$ balls are drawn with replacement, and $14$ are red.
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a) Give the point estimator of $p$ and its value.
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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 $E = z_{\alpha/2} \times \sqrt{\frac{\hat{p} \times (1 - \hat{p})}{n}}$.
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c) What is the relationship between this property and the law of large numbers?
Back to the beginning: Yes, it "counts": the mean of the $100$ responses, $\bar{x}=\frac{\sum x_i}{100}$, is the point estimate of the real spending $\mu$, and since $\bar{x}$ is an unbiased estimator, it is not systematically off.