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Example 1 The lengths of the sardines received by a certain cannery is normally distributed with mean 4.62 inches and a standard deviation 0.23 inch. What percentage of all these sardines is between 4.35 and 4.85 inches long? Sure, here's a concise version: Example 2 A baker knows that the daily de...

Example 1 The lengths of the sardines received by a certain cannery is normally distributed with mean 4.62 inches and a standard deviation 0.23 inch. What percentage of all these sardines is between 4.35 and 4.85 inches long? Sure, here's a concise version: Example 2 A baker knows that the daily demand for apple pies is a random variable which follows the normal distri- bution with mean 43.3 pies and standard deviation 4.6 pics. Find the demand which has probability 5% of being exceeded Example 3 Suppose that the height of UCLA female students has normal distribution with mean 62 inches and standard deviation 8 inches. a. Find the height below which is the shortest 30% of the female students. b. Find the height above which is the tallest 5% of the female students. Example 5 To avoid accusatious of sexism in a college class equally populated by inale and female students, the professor flips a fnir coin to decide whether to call upon a male or female student to answer a question directed to the class. The professor will call upon a female student if a tails occurs. Suppose the professor does this 1000 times during the semester. a. What is the probability that he calls upon a female student at least 530 times? b. What is the probability that he calls upon a female student at most 480 times? c. What is the probability that he calls upon a female student exactly 510 times? Example 6 MENSA is an organization whose members possess IQs in the top 2% of the population. a. If IQs are normally distributed, with mean 100 and a standard deviation of 16. what is the minimum IQ required for admission to MENSA? b. If three individuals are chosen at random from the general population what is the probability that all three satisfy the minimum requirement for MENSA? Example 7 A manufacturing process produces semiconductor chips with a known failure rate 6.3%. Assume that chip failures are independent of one another. You will be producing 2000 chips tomorrow. a. Find the expected number of defective chips produced. b. Find the standard deviation of the number of defective chips. e. Find the probability (approximate) that you will EXERCISE 8  Suppose that the height (V) in inches, of a 25-year-old man is a normal random variable with mean = 70 inches. If P(X79) = 0.025 what is the standard deviation of this random normal variable? EXERCISE 9 Suppose that the weight (X) in ponds, of a 10-year-old man is a normal random variable with standard deviation = 20 pounds. If 5% of this population is heavier than 214 pounds what is the mean a of this distribution? Problem 10 At Hertz ketel mp factory the amounts which go into bottles of ketelmp are supposed to be normally distributed with mean 36 oz. and standard deviation 0.1 oz. Once every 30 minutes a bottle is selected from the production line, and its contents are noted precisely. If the amount of the bottle goes below 30.8.or above 36.2 oz. then the bottle will be declared out of control. a. If the process is in control, meaning μ= 36 oz. and σ = 0.1 oz., find the probability that a bottle will be declared out of control. b. In the situation of (a), find the probability that the number of bottles found out of control in an eight-hour day (16 inspections) will be zero. c. In the situation of (a), find the probability that the number of bottles found out of control in an eight-hour day (16 inspections) will be exactly one. d. If the process shifts so that μ= 37 oz and σ = 0.4 oz. hud the probability that a bottle will be declared out of control. Problem 11 Suppose that a binary message -either 0 or 1- must be trasmitted by wire from location A to location B. However, the data sent over the wire are subject to a channel noise disturbance, so to reduce the possibilty of error. the value 2 is sent over the wire when the message is 1 and the value -2 is sent when the message is 0. If r, r = ±2. is the value sent from location A, then R. the value received at location B, is given by Rr+N. where V is the chamuel noise disturbance. When the message is received at location B the receiver decodes it according to the following rule: If R≥ 0.5. then 1 is concluded If R < 0.5. then 0 is concluded If the channel noise follows the standard normal distribution compute the probability that the message will be wrong when decoded. Problem 1 The chickens of the Ornithes farm are processed when they are 20 weeks old. The distribution of their weights is normal with mean 3.8 lb, and standard deviation 0.6 lb. The farm has created three categories for these chickens according to their weight: petite (weight less than 3.5 lb), standard (weight between 3.5 lb and 4.9 lb), and big (weight above4.9 lb). a. What proportion of these chickens will be in each category? Show these proportions on the normal distribution graph. b. Find the 60th percentile of the distribution of the weight. In other words find e such that P(X < c) = 0.60. c. Suppose that 5 chickens are selected at random. What is the probability that 3 of them will be petite? Problem 2 A television cable company receives numerous phone calls throughout the day from customers reporting service troubles and from would-be subscribers to the cable network. Most of these callers are put "on hold" until a company operator is free to help them. The company has determined that the length of time a caller is on hold is normally distributed with a mean of 3.1 minutes and a standard deviation 0.9 minutes. Company experts have decided that if as many as 5% of the callers are put on hold for 4.8 minutes or longer, more operators should be hired. a. What proportion of the company's callers are put on hold for more than 4.8 minutes? Should the company hire more operators? Show these probabilities on a sketch of the normal curve. b. At another cable company (length of time a caller is on hold follows the same distri- bution as before), 2.5% of the callers are put on hold for longer than z minutes. Find the value of r. and show this on a sketch of the normal curve Problem 3 Answer the following questions: a. Suppose that the height (X) in inches, of a 25-year-old man is a normal random variable with mean = 70 inches. If P(X > 79) = 0.025 what is the standard deviation of this random normal variable? b. Suppose that the weight (X) in pounds, of a 40-year-old man is a normal random variable with standard deviation σ = 20 pounds. If 5% of this population weigh less than 160 pounds what is the mean p of this distribution? c. Find an interval that covers the middle 95% of X ~ N(64.8) Problem 4 A bag of cookies is underweight if it weighs less than 500 grams. The filling process dispenses cookies with weight that follows the normal distribution with mean 510 grams and standard deviation 4 grams. a. What is the probability that a randomly selected bag is underweight? b. If you randomly select 5 bags, what is the probability that exactly 2 of them will be underweight? Problem 5 Answer the following questions: a. Suppose that X follows the normal distribution with mean µ = 5. If P(X > 9) = 0.2 find the variance of X. b. Let X be a normal random variable with mean = 12 and standard deviation o = 2. Find the 10th percentile of this distribution. c. The weight X of water melons is normally distributed with mean µ = 10 pounds and standard deviation = 2 pounds. Find e such that P(X > c) = 0.60. d. The montly return of a particular stock follows the normal distribution with mean 0.02 and standard deviation 0.1. Find the 85th percentile of this distribution. e. Find the probability that the monthly return of the stock in question (b) will be larger that 0.2. f. Find the probability that in one year (12 months), the return of the stock in question (e) will be larger than 0.2 on exactly 4 months. Assume that the returns are independent from month to month. g. The annual rainfall X (in inches) at a certain region is normally distributed with mean 1 = 40 pounds and standard deviation = 4. What is the probability that starting with this year, it will take more than 10 years before a year occurs having a rainfall of over 50 inches? h. Let X ~ N(100.20). Find P(X > 70\X < 90). Problem 6 The diameters of apples from the Milo Farm follow the normal distribution with mean 3 inches and standard deviation 0.3 inch. Apples can be size-sorted by being made to roll over a mesh screens. First the apples are rolled over a screen with mesh size 2.5 inches. This separates out all the apples with diameters < 2.5 inches. Second, the remaining apples are rolled over a screen with mash size 3.2 inches. Find the proportion of apples with diameters < 2.5 inches. between 2.5 and 3.2 inches, and greater than 3.2 inches. Use only SOCR to find the answers and print the appropriate snapshots.