The data set mentioned in Exercise 1 also includes these...

The data set mentioned in Exercise 1 also includes these third-grade IQ observations for males: $\begin{array}{lll}{117} & {103} & {121} \\ {149} & {125} & {131}\end{array}$ $\begin{array}{lll}{112} & {120} & {132} \\ {136} & {107} & {108}\end{array}$ $\begin{array}{lll}{113} & {117} & {132} \\ {113} & {136} & {114}\end{array}$ and females: $\begin{array}{lll}{114} & {102} & {113} \\ {114} & {109} & {102}\end{array}$ $\begin{array}{ll}{131} & {124} \\ {114} & {127}\end{array}$ $\begin{array}{ll}{117} & {120} \\ {127} & {103}\end{array}$ 90 Prior to obtaining data, denote the male values by $X_{1}, \ldots, X_{m}$ and the female values by $Y_{1}, \ldots, Y_{n}$ Suppose that the $X_{i}$ s constitute a random sample from a distribution with mean $\mu_{1}$ and standard deviation $\sigma_{1}$ and that the $Y_{i}$ s form a random sample (independent of the $X_{i}$ s ) from another distribution with mean $u_{2}$ and standard deviation $\sigma_{2}$ . (a) Show that $\overline{X}-\overline{Y}$ is an unbiased estimator of $\mu_{1}-\mu_{2}$ . Then calculate the estimate for the given data. (b) Use rules of variance from Chap. 4 to obtain an expression for the standard error of the estimator in (a), and then compute the estimated standard error. (c) Calculate a point estimate of the ratio $\sigma_{1} / \sigma_{2}$ of the two standard deviations. (d) Suppose one male third-grader and one female third-grader are randomly selected. Calculate a point estimate of the variance of the difference $X-Y$ between their IQs.

The data set mentioned in Exercise 1 also includes these third-grade IQ observations for males:
$\begin{array}{lll}{117} & {103} & {121} \\ {149} & {125} & {131}\end{array}$
$\begin{array}{lll}{112} & {120} & {132} \\ {136} & {107} & {108}\end{array}$
$\begin{array}{lll}{113} & {117} & {132} \\ {113} & {136} & {114}\end{array}$
and females:
$\begin{array}{lll}{114} & {102} & {113} \\ {114} & {109} & {102}\end{array}$
$\begin{array}{ll}{131} & {124} \\ {114} & {127}\end{array}$
$\begin{array}{ll}{117} & {120} \\ {127} & {103}\end{array}$ 90
Prior to obtaining data, denote the male values by $X_{1}, \ldots, X_{m}$ and the female values by $Y_{1}, \ldots, Y_{n}$
Suppose that the $X_{i}$ s constitute a random sample from a distribution with mean $\mu_{1}$ and standard
deviation $\sigma_{1}$ and that the $Y_{i}$ s form a random sample (independent of the $X_{i}$ s ) from another
distribution with mean $u_{2}$ and standard deviation $\sigma_{2}$ .
(a) Show that $\overline{X}-\overline{Y}$ is an unbiased estimator of $\mu_{1}-\mu_{2}$ . Then calculate the estimate for the
given data.
(b) Use rules of variance from Chap. 4 to obtain an expression for the standard error of the
estimator in (a), and then compute the estimated standard error.
(c) Calculate a point estimate of the ratio $\sigma_{1} / \sigma_{2}$ of the two standard deviations.
(d) Suppose one male third-grader and one female third-grader are randomly selected. Calculate
a point estimate of the variance of the difference $X-Y$ between their IQs.

Key Concepts

Unbiased Estimator

An unbiased estimator is a statistical estimate whose expected value equals the true value of the parameter being estimated. This property ensures that, on average, the estimator neither overestimates nor underestimates the parameter, making it a foundational concept in point estimation and inferential statistics.

Difference in Sample Means

In comparing two populations, the difference between the sample means is commonly used to estimate the difference between the population means. Because the sample means are unbiased estimators of their respective population means, their difference is also an unbiased estimator of the difference between the population means.

Variance Rules for Independent Random Variables

When dealing with sums or differences of independent random variables, the variance of the linear combination is the sum of the variances of the individual random variables (weighted appropriately). This principle is crucial when calculating the standard error of estimators that involve differences of sample means or individual observations.

Standard Error

The standard error quantifies the variability of an estimator by measuring the standard deviation of its sampling distribution. It provides insight into the precision of the estimator and is essential for constructing confidence intervals and performing hypothesis tests.

Point Estimation

Point estimation involves using sample data to compute a single best estimate of an unknown population parameter. It serves as a critical step in inferential statistics, where these estimates are then used to make broader claims about the population.

Estimating Ratios of Standard Deviations

Estimating the ratio of two standard deviations involves using the corresponding sample standard deviations as surrogates for their unknown population counterparts. This technique allows for direct comparison of the spread or variability in two different populations.

Variance of the Difference of Independent Observations

When evaluating the difference between two independent observations (each from different populations), the variance of the difference is obtained by summing the variances of the individual observations. This concept is important for understanding and quantifying the uncertainty in such comparisons.

Transcript

00:01 In problem 4, we have some data on that grade iq observations for males and females.

00:09 And so we are also given that the male values are denoted by x1, x2 through to xm, and the female values are denoted by y1, y2, all through to yn.

00:25 This means that the sample size for males is m and the sample size for females is n.

00:35 We also told that the males, the samples that were obtained from males come from a distribution that has a mean of mu 1 and the standard deviation sigma 1.

00:49 While for females the distribution has a mean mu 2 and the standard deviation sigma 2.

00:58 So you're going to be going to use this data and this information to solve a few problems and the first problem we're going to be showing that x bar minus y bar is an unbiased estimator of mu 1 minus mu 2.

01:18 After that we're going to calculate the estimate for using the given data.

01:22 So the first thing we need to do for part a, we need to show that x bar, x bar minus y bar is an unbiased estimator of mu 1 minus mu 2.

01:53 To show that it is an estimator an unbiased estimator then we have to show that the expectation of x x bar minus y bar will be equal to mu one minus mu two.

02:13 Once we show that, then we have completed the proof.

02:18 So let's begin solving the left -hand side of the equation, the expectation of x bar minus y bar.

02:27 So the expectation of x bar minus y bar can be solved as follows.

02:32 We have to write the expectation of x bar minus y bar in this form, it's going to be equal to the expectation of x bar minus the expectation of y bar.

02:50 Because the expectation is a linear operator.

02:56 Next, we need to write x bar in the form that, in the formula, in the form that it has been, can be worked out.

03:06 So x bar is the mean of these sample values for males and y bar is the mean for the sample values for females.

03:14 So it means that the expectation of x bar can be written as expectation of 1 over m, which is the m stands for the sample size, multiplied by the sum of x's.

03:32 So that means x1, x2, all the way to xm, minus the expectation of y bar.

03:50 And the expectation of y bar is expectation of the mean of y.

03:55 And the mean of y is given by 1 over n, which is the sample size, multiplied by the sum of the y values.

04:04 That is y1 plus y2 all the way until we get to y n.

04:18 Next, we can factor out the constant 1 over m so that we have 1 over m expectation of x1 added all through until xm.

04:41 Then you do the same thing and factor out the factor out the constant 1 over n so that we have 1 over n expectation of y 1 plus all the values of y until the last one y n okay and now that we have the the sum of values of x we can solve the expectation of the sum as follows, it's going to be equal to 1 over m.

05:20 And you can put the brackets here.

05:23 Expectation of x1 plus the expectation of x2 all the way through until you get to the expectation of xm minus 1 over n multiplied by the expectation of y1 plus the expectation of y1 plus the expectation of y2.

05:54 Y3 all the way until we get to the expectation of y n good and since we know the distribution where the values of x and y come from then it has a mean of y mu 1 for x and a mean of mu 2 for y then we can conclude that each of these expectations will be mu 1 for for males and each of the expectations will be mu 2 for female so it's going to be 1 over m multiplied by mu 1 added to itself all the way through until we get to mu 1 so that means we're adding mu 1 to itself m times because the sample size is m in the same way we go on end give this as 1 over n multiplied by the expectation of y1 is mu 2 which will be be added to itself and n times until we get to the last value.

07:06 So all this will be n times.

07:13 So when we simplify that, it's going to be 1 over m multiplied by m mu 1.

07:22 If you add mu 1 to itself and when you add all these values of mu2, n times, we'll get n, mu2.

07:40 Now we can cancel out the m, the m.

07:47 Okay.

07:49 So when you cancel that out, you will get mu 1 and we can minus, cancel out the n and the n here, and we'll get mu2.

08:00 So the expectation of x bar minus y bar is mu1 minus mu 2.

08:10 And because of that fact, we can conclude that the x bar minus y bar is an unbiased estimator of mu1 minus mu2.

08:22 Next, we're going to calculate the estimate for the given data.

08:26 So we need to calculate the estimate of the difference between the means.

08:32 So that means we're going to be getting mu...

08:37 Sorry, we're going to be getting mu 1 hut minus.....