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What is a density curve, and why are such curves important?

11. Answer true or false to each statement. Explain your answers.

a. Two normal distributions that have the same mean are centered at the same place, regardless of the relationship between their standard deviations.

b. Two normal distributions that have the same standard deviation have the same spread, regardless of the relationship between their means.

For the standard normal curve, find the z- score(s)

a. that has area 0.30 to its left.

b. that has area 0.10 to its right.

c. $z_{0.025},z_{0.05},z_{0.01}and{z}_{0.005}.$

d. that divide the area under the curve into a middle 0.99 area and two outside 0.005 areas

Dispensing Coffee. A coffee machine is supposed to dispense 6 fluid ounces (fi oz) of coffee into a paper cup. In reality, the amounts dispensed vary from cup to cup. In fact, the amount dispensed. in fi oz. is a variable with density curve y=2 for 5.75< x < 6.25, and y = 0 otherwise.

a. Graph the density curve of this variable.

b. Show that the area under this density curve to the left of any number x between 5.75 and 6 .25 equals 2x-11.5. What percentage of cups dispensed by this machine contain

c. less than 6 fi oz?

d. between 5.9 and 6.1 11 oz

e. at least 5.8 fi oz.?

The area under a density curve that lies to the left of 60 is $0.364$. What percentage of all possible observations of the variable are

a. Less than 60?

b. At least 60?

From the paper "Effects of Chronic Nitrate Exposure on Gonad Growth in Green Sea Urchin Strongylocentrotus droebachiensis" by S. Siikavuopio et al., we found that weights of adult green sea urchins are normally distributed with mean $52g$and standard deviation $17.2g$.

Part (a): Find the percentage of adult green sea urchins with weights between $50g$and $60g$.

Part (b): Obtain the percentage of adult green sea urchins with weight above $40g$.

Part (c): Determine and interpret the $90th$percentile for the weights.

Part (d): Find and interpret the $6th$decile for the weights.

Explain in detail what a normal probability plot is and how it is used to assess the normality of a variable.

Explain how to obtain normal scores from Table III in Appendix $A$ when a sample contains equal observations.

Desert Samaritan Hospital in Mesa, Arizona, keeps records of emergency room traffic. Those records reveal that the times between arriving patients have a special type of reverse-J-shaped distribution called an exponential distribution. The records also show that the mean time between arriving patients is $8.7$8 minutes.

a. Use the technology of your choice to simulate four random samples of$75$ interarrival times each.

b. Obtain a normal probability plot of each sample in part (a).

c. Are the normal probability plots in part (b) what you expected? Explain your answer.

Sketch the normal distribution with

$$\begin{gathered} \left(a\right)\mathrm{μ}=3;\mathrm{σ}=3 \\ \left(b\right)\mathrm{μ}=1;\mathrm{σ}=3 \\ \left(c\right)\mathrm{μ}=3;\mathrm{σ}=1 \end{gathered}$$