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Suppose x is a normally distributed random variable with $\mathrm{渭}$= 11 and $\mathrm{蟽}$= 2. Find each of the following:

a)$\mathbf{P}(\mathbf{10}鈮�\mathbf{蠂}鈮�\mathbf{12})$

b) $\mathbf{P}(\mathbf{6}鈮�\mathbf{蠂}鈮�\mathbf{10})$

c)$\mathbf{P}(\mathbf{13}鈮�\mathbf{蠂}鈮�\mathbf{16})$

d)$\mathbf{P}(\mathbf{7}\mathbf{.}\mathbf{8}鈮�\mathbf{蠂}鈮�\mathbf{12}\mathbf{.}\mathbf{6})$

e)$\mathbf{P}(\mathbf{蠂}鈮�\mathbf{13}\mathbf{.}\mathbf{24})$

f)$\mathbf{P}(\mathbf{蠂}鈮�\mathbf{7}\mathbf{.}\mathbf{62})$

Suppose $\mathbf{蠂}$ is a normally distributed random variable with $\mathbf{渭}\mathbf{=}\mathbf{50}$ and $\mathbf{蟽}\mathbf{=}\mathbf{3}$ . Find a value of the random variable, call it ${\mathbf{蠂}}_{\mathbf{0}}$ , such that

a)$\mathbf{P}(\mathbf{蠂}鈮�{\mathbf{蠂}}_0)\mathbf{=}\mathbf{.}\mathbf{8413}$

b)$\mathbf{P}(\mathbf{蠂}>{\mathbf{蠂}}_0)\mathbf{=}\mathbf{.}\mathbf{025}$

c)$\mathbf{P}(\mathbf{蠂}>{\mathbf{蠂}}_0)\mathbf{=}\mathbf{.}\mathbf{95}$

d)$\mathbf{P}(\mathbf{41}鈮�\mathbf{蠂}鈮�{\mathbf{蠂}}_0)\mathbf{=}\mathbf{.}\mathbf{8630}$

e) 10% of the values of role="math" localid="1652160513072" $\mathbf{蠂}$are less thanrole="math" localid="1652160519976" ${\mathbf{蠂}}_{\mathbf{0}}$

f)1% of the values of$\mathit{蠂}$ are greater than${\mathbf{蠂}}_{\mathbf{0}}$

Suppose x is a binomial random variable with p = .4 and n = 25.

a. Would it be appropriate to approximate the probability distribution of x with a normal distribution? Explain.

b. Assuming that a normal distribution provides an adequate approximation to the distribution of x, what are the mean and variance of the approximating normal distribution?

c. Use Table I in Appendix D to find the exact value of $\mathbf{P}\mathbf{(}\mathit{x}\mathbf{鈮�}\mathbf{9}\mathbf{)}$.

d. Use the normal approximation to find $\mathbf{P}\mathbf{(}\mathit{x}\mathbf{鈮�}\mathbf{9}\mathbf{)}$.

Variable life insurance return rates. With a variable life insurance policy, the rate of return on the investment (i.e., the death benefit) varies from year to year. A study of these variable return rates was published in International Journal of Statistical Distributions (Vol. 1, 2015). A transformedratio of the return rates (x) for two consecutive years was shown to have a normal distribution, with $\mathit{渭}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{5}$ and role="math" localid="1660283206727" $\mathit{蟽}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{2}$. Use the standard normal table or statistical software to find the following probabilities.

a.$\mathbf{P}\mathbf{(}\mathbf{1}\mathbf{.}\mathbf{3}\mathbf{<}\mathit{x}\mathbf{<}\mathbf{1}\mathbf{.}\mathbf{6}\mathbf{)}$

b. $\mathbf{P}\mathbf{(}\mathit{x}\mathbf{>}\mathbf{1}\mathbf{.}\mathbf{4}\mathbf{)}$

c. $\mathbf{P}\mathbf{(}\mathit{x}\mathbf{<}\mathbf{1}\mathbf{.}\mathbf{5}\mathbf{)}$

Hotels鈥� use of ecolabels. Refer to the Journal of Vacation Marketing (January 2016) study of travelers鈥� familiarity with ecolabels used by hotels, Exercise 2.64 (p. 104). Recall that adult travelers were shown a list of 6 different ecolabels, and asked, 鈥淪uppose the response is measured on a continuous scale from 10 (not familiar at all) to 50 (very familiar).鈥� The mean and standard deviation for the Energy Star ecolabel are 44 and 1.5, respectively. Assume the distribution of the responses is approximately normally distributed.

a. Find the probability that a response to Energy Star exceeds 43.

b. Find the probability that a response to Energy Star falls between 42 and 45.

c. If you observe a response of 35 to an ecolabel, do you think it is likely that the ecolabel was Energy Star? Explain.