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Chapter 8: Problem 136

The sample mean \(\bar{Y}\) is a good point estimator of the population mean \(\mu\). It can also be used to predict a future value of \(Y\) independently selected from the population. Assume that you have a sample mean \(\bar{Y}\) and variance \(S^{2}\) based on a random sample of \(n\) measurements from a normal population. Use Student's \(t\) to form a pivotal quantity to find a prediction interval for some new value of \(Y-\) say, \(X_{p}-\) to be observed in the future. [Hint: Start with the quantity \(Y_{p}-\bar{Y} .\) ] Notice the terminology: Parameters are estimated; values of random variables are predicted.

Short Answer

Expert verified
The prediction interval for the new value \(Y_p\) is \(\bar{Y} \pm t_{\alpha/2, n-1} S\sqrt{1 + \frac{1}{n}}\).

Step by step solution

01

Understand the Problem

We need to find a prediction interval for a new observation, \(Y_p\), based on a normal population from which we've taken a sample. We have the sample mean \(\bar{Y}\) and sample variance \(S^2\). The problem requires us to use Student's \(t\) distribution because the population variance is unknown.
02

Determine Pivotal Quantity

A pivotal quantity is a function of observed data and unknown parameters that has a known distribution. Here, the hint suggests starting with the expression \(Y_p - \bar{Y}\). Consider the pivotal quantity \(\frac{Y_p - \bar{Y}}{S\sqrt{1 + \frac{1}{n}}}\). This quantity is \(t\) distributed with \(n - 1\) degrees of freedom because \(Y_p\), the new observation, is independent of the sample \(\bar{Y}\).
03

Formulate Prediction Interval

Using the pivotal quantity, we set up the interval as \(\bar{Y} \pm t_{\alpha/2, n-1} S\sqrt{1 + \frac{1}{n}}\). Here, \(t_{\alpha/2, n-1}\) is the critical value of the \(t\)-distribution for the desired confidence level with \(n-1\) degrees of freedom.
04

Final Prediction Interval Expression

Hence, the prediction interval for the new observation \(Y_p\) is \(\left( \bar{Y} - t_{\alpha/2, n-1} S\sqrt{1 + \frac{1}{n}}, \bar{Y} + t_{\alpha/2, n-1} S\sqrt{1 + \frac{1}{n}} \right)\). This interval accounts for the additional variability when predicting a new individual observation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sample Mean
The sample mean, denoted \(\bar{Y}\), is a simple yet powerful tool in statistics. It represents the average of a set of measurements taken from a population. Specifically, when you compute the sample mean, you're summing up all the data points within your sample and then dividing by the number of data points. This gives you a single number that serves as an estimate of the population mean. It's called a "point estimator" because it provides one specific value to predict the overall mean of the population.
  • The sample mean is used as the best estimate of the actual population mean \(\mu\).
  • It's particularly useful when the population mean is not directly calculable due to constraints such as time or practicality.
  • Despite being an estimate, the sample mean is quite reliable, especially when the sample size \(n\) is large.
Understanding the sample mean is crucial because it acts as the foundation for more complex statistical analyses, like prediction intervals.
Student's t-distribution
The Student's t-distribution plays a vital role when we work with small sample sizes or when the population variance is unknown. Unlike the normal distribution, which is applicable to large samples, the t-distribution is more adaptable for smaller ones. It is crucial because as sample size decreases, the distribution becomes wider, reflecting greater uncertainty.
  • The t-distribution is symmetric and bell-shaped, somewhat like the normal distribution but with heavier tails. This shape accounts for higher variability in small samples.
  • This distribution depends on degrees of freedom, calculated as \(n-1\) for a given sample size \(n\).
  • It's widely used to construct confidence intervals and test hypotheses where the population standard deviation is not known.
In the context of our exercise, we use the t-distribution because we are predicting a future observation and we don鈥檛 know the population variance.
Pivotal Quantity
The concept of a pivotal quantity is somewhat esoteric but extremely useful in statistical inference. A pivotal quantity is a function of both sample data and unknown parameters that has a known probability distribution.
  • In the exercise, the pivotal quantity \(\frac{Y_p - \bar{Y}}{S\sqrt{1 + \frac{1}{n}}}\) is introduced. This expression is essential because it follows a known distribution - specifically, the t-distribution.
  • By manipulating the distribution of this quantity, we can derive prediction intervals for future observations.
  • Given that this pivotal quantity follows a t-distribution, it allows for the calculation of interval estimates even when parameters like the variance of the population are not known.
By establishing a pivotal quantity, statisticians can more accurately predict future events or analyze hypotheses, even with limited data sets.
Normal Population
When discussing statistics, a normal population is one where the variable of interest follows a normal distribution鈥攐ften depicted as a bell curve. This common distribution is foundational in statistical techniques because of its predictable properties.
  • A normal distribution is defined by two parameters: the mean \(\mu\) and the variance \(\sigma^2\).
  • Its symmetrical bell shape means that most of the data points lie around the mean, with fewer situations of extreme values.
  • Many natural phenomena and characteristics like height, test scores, or measurement errors follow a normal distribution.
In the exercise, the assumption of a normal population makes it possible to use statistical tools like the sample mean and pivotal quantities to make predictions and construct intervals, due to the known properties of normal distributions.

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Most popular questions from this chapter

Suppose that we obtain independent samples of sizes \(n_{1}\) and \(n_{2}\) from two normal populations with equal variances. Use the appropriate pivotal quantity from Section 8.8 to derive a \(100(1-\alpha) \%\) upper confidence bound for \(\mu_{1}-\mu_{2}\)

The number of breakdowns per week for a type of minicomputer is a random variable \(Y\) with a Poisson distribution and mean \(\lambda\). A random sample \(Y_{1}, Y_{2}, \ldots, Y_{n}\) of observations on the weekly number of breakdowns is available. a. Suggest an unbiased estimator for \(\lambda\). b. The weekly cost of repairing these breakdowns is \(C=3 Y+Y^{2} .\) Show that \(E(C)=4 \lambda+\lambda^{2}\) c. Find a function of \(Y_{1}, Y_{2}, \ldots, Y_{n}\) that is an unbiased estimator of \(E(C)\). [Hint: Use what you know about \(\left.\bar{Y} \text { and }(\bar{Y})^{2} .\right]\)

Suppose that \(E\left(\hat{\theta}_{1}\right)=E\left(\hat{\theta}_{2}\right)=\theta, V\left(\hat{\theta}_{1}\right)=\sigma_{1}^{2},\) and \(V\left(\hat{\theta}_{2}\right)=\sigma_{2}^{2} .\) Consider the estimator \(\hat{\theta}_{3}=a \hat{\theta}_{1}+(1-a) \hat{\theta}_{2}\) a. Show that \(\hat{\theta}_{3}\) is an unbiased estimator for \(\theta\). b. If \(\hat{\theta}_{1}\) and \(\hat{\theta}_{2}\) are independent, how should the constant \(a\) be chosen in order to minimize the variance of \(\hat{\theta}_{3} ?\)

We noted in Section 8.3 that if $$S^{\prime 2}=\frac{\sum_{i=1}^{n}\left(Y_{i}-\bar{Y}\right)^{2}}{n} \text { and } S^{2}=\frac{\sum_{i=1}^{n}\left(Y_{i}-\bar{Y}\right)^{2}}{n-1}$$ then \(S^{\prime 2}\) is a biased estimator of \(\sigma^{2}\), but \(S^{2}\) is an unbiased estimator of the same parameter. If we sample from a normal population, a. find \(V\left(S^{\prime 2}\right)\) b. show that \(V\left(S^{2}\right)>V\left(S^{\prime 2}\right)\)

The number of persons coming through a blood bank until the first person with type \(A\) blood is found is a random variable \(Y\) with a geometric distribution. If \(p\) denotes the probability that any one randomly selected person will possess type \(A\) blood, then \(E(Y)=1 / p\) and \(V(Y)=(1-p) / p^{2}\) a. Find a function of \(Y\) that is an unbiased estimator of \(V(Y)\). b. Suggest how to form a 2 -standard-error bound on the error of estimation when \(Y\) is used to estimate \(1 / p\).
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