What does the sampling distribution of the sample variance look like if we sample from a population with an approximately normal distribution? Find out using the applet Sampling Distribution of the Variance (Mound Shaped Population) (at academic.cengage.com/statistics/wackerly) to complete the following. In the previous exercises in this section, you obtained simulated sampling distributions for the sample mean. All these sampling distributions were well approximated (for large sample sizes) by a normal distribution. Although the distribution that you obtained is mound-shaped, does the sampling distribution of the sample variance seem to be symmetric (like the normal distribution)?

Why do two normal distributions that have equal standard deviations have the same shape?

What does the sampling distribution of the sample variance look like if we sample from a population with an approximately normal distribution? Find out using the applet Sampling Distribution of the Variance (Mound Shaped Population) (at academic.cengage.com/statistics/wackerly) to complete the following. Click the button “Toggle Theory” to overlay the theoretical density function for the sampling distribution of the variance of a sample of size 5 from a normally distributed population. Does the theoretical density provide a reasonable approximation to the values represented in the histogram?

What does the sampling distribution of the sample variance look like if we sample from a population with an approximately normal distribution? Find out using the applet Sampling Distribution of the Variance (Mound Shaped Population) (at academic.cengage.com/statistics/wackerly) to complete the following. Click the button “Next Obs” to take a sample of size 1 from the population with distribution represented by the top histogram. The value obtained is plotted on the middle histogram. Click four more times to complete a sample of size 5. The value of the sample variance is computed and given above the middle histogram. Is the value of the sample variance equal to the value of the population variance? Does this surprise you?

Suppose that X1,.............., Xn is a random sample from the exponential distribution with parameter $\theta$. Find the c.d.f. for the sampling distribution of the M.L.E. of $\theta$.

In the article “Reproductive Ecology and Cub Survival of Florida Black Bears”, E. Garrison et al. investigated cub survival rates of the Florida black bear. Through the observation of female Florida black bears and their litters, the following estimate is given for the probability distribution of the number of young per litter for female Florida black bears.

$$\begin{array}{l|cccc} \hline \text { Litter size } & 1 & 2 & 3 & 4 \\ \hline \text { Probability } & 0.212 & 0.606 & 0.152 & 0.030 \\ \hline \end{array}$$

a. Explain Why the probabilities in the table provide only an estimate of the true probabilities. b. Estimate the probability that a Florida black bear has a litter of either two or three cubs.

The sampling distribution describe the distribution of what?

Assume that the 'n' random variables X1,........., Xn form a random sample from a continuous distribution for which the p.d.f. is 'f'. Determine the probability that at least 'k' of these 'n' random variables will lie in a specified interval $a \leq x \leq b$.

Adult female black bears have masses from about 50 kilograms to 125 kilograms. Adult male black bears have masses from about 60 kilograms to 250 kilograms. Researchers measured the masses of five adult female black bears in their natural habitat. You can see their results in the table.

Adult Female

Black Bear Masses

Which mass is anomalous data?

What is the formula for the standard error of the normal approximation to the $\hat{p}$ distribution and what is the mean of the $\hat{p}$ distribution?

Transactions to a computer database are either new items or changes to previous items. The addition of an item can be completed in less than 100 milliseconds 90% of the time, but only 20% of changes to a previous item can be completed in less than this time. If 30% of transactions are changes, what is the probability that a transaction can be completed in less than 100 milliseconds?

Refer to the international Journal of Retail and Distribution Management (Vol. 39, 2011) survey of Black Friday shoppers. One question was "How many hours do you usually spend shopping on Black Friday?" How many Black Friday shoppers should be included in a sample designed to estimate (with 95% confidence) the average number of hours spent shopping on Black Friday if you want the estimate to deviate no more than .5 hour from the true mean?

Are their any outliers? Is the relationship (ignoring the outliers) roughly linear in form? Still ignoring outliers, how strong would you say the relationship between calories and sodium is?

Solve the linear system using substitution.

$$\left. \begin{array} { l } { x - 9 = 0.5 y } \\ { 2.2 x - 3.1 y = - 0.2 } \end{array} \right.$$