91Ó°ÊÓ

Assume that the random variable \(X\) is normally distributed, with mean \(\mu=50\) and standard deviation \(\sigma=7 .\) Compute the following probabilities. Be sure to draw a normal curve with the area corresponding to the probability shaded. \(P(56 \leq X<66)\)

The birth weights of full-term babies are normally distributed with mean \(\mu=3400\) grams and \(\sigma=505\) grams. Source: Based on data obtained from the National Vital Statistics Report, Vol. \(48,\) No. 3 (a) Draw a normal curve with the parameters labeled. (b) Shade the region that represents the proportion of full-term babies who weigh more than 4410 grams. (c) Suppose the area under the normal curve to the right of \(x=4410\) is \(0.0228 .\) Provide two interpretations of this result.

Assume that the random variable \(X\) is normally distributed, with mean \(\mu=50\) and standard deviation \(\sigma=7\). Find each indicated percentile for \(X\) The 9 th percentile

Assume that the random variable \(X\) is normally distributed, with mean \(\mu=50\) and standard deviation \(\sigma=7\). Find each indicated percentile for \(X\) The 90th percentile

The mean incubation time of fertilized chicken eggs kept at \(100.5^{\circ} \mathrm{F}\) in a still-air incubator is 21 days. Suppose that the incubation times are approximately normally distributed with a standard deviation of 1 day. Source: University of Illinois Extension (a) Draw a normal model that describes egg incubation times of fertilized chicken eggs. (b) Find and interpret the probability that a randomly selected fertilized chicken egg hatches in less than 20 days. (c) Find and interpret the probability that a randomly selected fertilized chicken egg takes over 22 days to hatch. (d) Find and interpret the probability that a randomly selected fertilized chicken egg hatches between 19 and 21 days. (e) Would it be unusual for an egg to hatch in less than 18 days? Why?

The reading speed of sixth-grade students is approximately normal, with a mean speed of 125 words per minute and a standard deviation of 24 words per minute. (a) Draw a normal model that describes the reading speed of sixth-grade students. (b) Find and interpret the probability that a randomly selected sixth-grade student reads less than 100 words per minute. (c) Find and interpret the probability that a randomly selected sixth-grade student reads more than 140 words per minute. (d) Find and interpret the probability that a randomly selected sixth-grade student reads between 110 and 130 words per minute. 0.3189 (e) Would it be unusual for a sixth grader to read more than 200 words per minute? Why?

The lengths of human pregnancies are approximately normally distributed, with mean \(\mu=266\) days and standard deviation \(\sigma=16\) days. (a) What proportion of pregnancies lasts more than 270 days? (b) What proportion of pregnancies lasts less than 250 days? (c) What proportion of pregnancies lasts between 240 and 280 days? (d) What is the probability that a randomly selected pregnancy lasts more than 280 days? (e) What is the probability that a randomly selected pregnancy lasts no more than 245 days? (f) A "very preterm" baby is one whose gestation period is less than 224 days. Are very preterm babies unusual?

General Electric manufactures a decorative Crystal Clear 60 -watt light bulb that it advertises will last 1500 hours. Suppose that the lifetimes of the light bulbs are approximately normally distributed, with a mean of 1550 hours and a standard deviation of 57 hours. (a) What proportion of the light bulbs will last less than the advertised time? (b) What proportion of the light bulbs will last more than 1650 hours? (c) What is the probability that a randomly selected GE Crystal Clear 60 -watt light bulb will last between 1625 and 1725 hours? (d) What is the probability that a randomly selected GE Crystal Clear 60 -watt light bulb will last longer than 1400 hours?

Steel rods are manufactured with a mean length of 25 centimeters \((\mathrm{cm}) .\) Because of variability in the manufacturing process, the lengths of the rods are approximately normally distributed, with a standard deviation of \(0.07 \mathrm{~cm} .\) (a) What proportion of rods has a length less than \(24.9 \mathrm{~cm} ?\) (b) Any rods that are shorter than \(24.85 \mathrm{~cm}\) or longer than \(25.15 \mathrm{~cm}\) are discarded. What proportion of rods will be discarded? (c) Using the results of part (b), if 5000 rods are manufactured in a day, how many should the plant manager expect to discard? (d) If an order comes in for 10,000 steel rods, how many rods should the plant manager manufacture if the order states that all rods must be between \(24.9 \mathrm{~cm}\) and \(25.1 \mathrm{~cm} ?\)

Ball bearings are manufactured with a mean diameter of 5 millimeters \((\mathrm{mm})\). Because of variability in the manufacturing process, the diameters of the ball bearings are approximately normally distributed, with a standard deviation of \(0.02 \mathrm{~mm}\) (a) What proportion of ball bearings has a diameter more than \(5.03 \mathrm{~mm} ?\) (b) Any ball bearings that have a diameter less than \(4.95 \mathrm{~mm}\) or greater than \(5.05 \mathrm{~mm}\) are discarded. What proportion of ball bearings will be discarded? (c) Using the results of part (b), if 30,000 ball bearings are manufactured in a day, how many should the plant manager expect to discard? (d) If an order comes in for 50,000 ball bearings, how many bearings should the plant manager manufacture if the order states that all ball bearings must be between \(4.97 \mathrm{~mm}\) and \(5.03 \mathrm{~mm} ?\)