STAT Unit 3

Find z such that 2.8% of the standard normal curve lies to the left of z. (Round your answer to two decimal places.) z =

-1.91

Let z be a random variable with a standard normal distribution. Find the indicated probability. (Round your answer to four decimal places.) P(z ≥ 1.99) =

.0233

Suppose 5% of the area under the standard normal curve lies to the left of z. Is z positive or negative?

Negative

Suppose 80% of the area under the standard normal curve lies to the right of z. Is z positive or negative?

Negative

Assume that x has a normal distribution with the specified mean and standard deviation. Find the indicated probability. (Round your answer to four decimal places.) μ = 20; σ = 4.2 P(x ≥ 30) =

.0087

Let z be a random variable with a standard normal distribution. Find the indicated probability. (Round your answer to four decimal places.) P(z ≤ −0.23) =

.4090

Let z be a random variable with a standard normal distribution. Find the indicated probability. (Enter your answer to four decimal places.) P(−2.05 ≤ z ≤ 1.09) =

.8419

Find z such that 62% of the standard normal curve lies to the left of z. (Round your answer to two decimal places.) z =

0.31

Find the z value such that 97% of the standard normal curve lies between −z and z. (Round your answer to two decimal places.) z =

2.17

Suppose x has a distribution with μ = 30 and σ = 29. (a) If a random sample of size n = 49 is drawn, find μx, σ x and P(30 ≤ x ≤ 32). (Round σx to two decimal places and the probability to four decimal places.) (b) If a random sample of size n = 74 is drawn, find μx, σ x and P(30 ≤ x ≤ 32). (Round σ x to two decimal places and the probability to four decimal places.) μx = σ x = (30 ≤ x ≤ 32) = (c) Why should you expect the probability of part (b) to be higher than that of part (a)? (Hint: Consider the standard deviations in parts (a) and (b).) The standard deviation of part (b) is _______ part (a) because of the ________ sample size. Therefore, the distribution about μx is _______

a) μx = 30 σ x = 4.14 P(30 ≤ x ≤ 32) = .1844 b) μx = 30 σ x = 3.37 (30 ≤ x ≤ 32) = .2224 c) smaller than ; larger ; ?

A normal distribution has μ = 24 and σ = 5. (a) Find the z score corresponding to x = 19. (b) Find the z score corresponding to x = 38. (c) Find the raw score corresponding to z = −2. (d) Find the raw score corresponding to z = 1.1.

a) -1.00 b) 2.80 c) 14.00 d) 29.50

Let x be a random variable that represents the weights in kilograms (kg) of healthy adult female deer (does) in December in a national park. Then x has a distribution that is approximately normal with mean μ = 69.0 kg and standard deviation σ = 6.5 kg. Suppose a doe that weighs less than 60 kg is considered undernourished. (a) What is the probability that a single doe captured (weighed and released) at random in December is undernourished? (Round your answer to four decimal places.) (b) If the park has about 2650 does, what number do you expect to be undernourished in December? (Round your answer to the nearest whole number.) (c) To estimate the health of the December doe population, park rangers use the rule that the average weight of n = 40 does should be more than 66 kg. If the average weight is less than 66 kg, it is thought that the entire population of does might be undernourished. What is the probability that the average weight x for a random sample of 40 does is less than 66 kg (assuming a healthy population)? (Round your answer to four decimal places.) (d) Compute the probability that x < 71 kg for 40 does (assume a healthy population). (Round your answer to four decimal places.) e)Suppose park rangers captured, weighed, and released 40 does in December, and the average weight was x = 71 kg. Do you think the doe population is undernourished or not? Explain.

a) .0838 b) 222 c) .0018 d) .9744 e) Since the sample average is above the mean, it is quite unlikely that the doe population is undernourished

Coal is carried from a mine in West Virginia to a power plant in New York in hopper cars on a long train. The automatic hopper car loader is set to put 61 tons of coal into each car. The actual weights of coal loaded into each car are normally distributed, with mean μ = 61 tons and standard deviation σ = 0.5 ton. (a) What is the probability that one car chosen at random will have less than 60.5 tons of coal? (Round your answer to four decimal places.) (b) What is the probability that 29 cars chosen at random will have a mean load weight x of less than 60.5 tons of coal? (Round your answer to four decimal places.) (c) Suppose the weight of coal in one car was less than 60.5 tons. Would that fact make you suspect that the loader had slipped out of adjustment? d) Suppose the weight of coal in 29 cars selected at random had an average x of less than 60.5 tons. Would that fact make you suspect that the loader had slipped out of adjustment? Why?

a) .1587 b) .0000 c) no d) Yes, the probability that this deviation is random is very small.

A person's blood glucose level and diabetes are closely related. Let x be a random variable measured in milligrams of glucose per deciliter (1/10 of a liter) of blood. Suppose that after a 12-hour fast, the random variable x will have a distribution that is approximately normal with mean μ = 85 and standard deviation σ = 20. Note: After 50 years of age, both the mean and standard deviation tend to increase. For an adult (under 50) after a 12-hour fast, find the following probabilities. (Round your answers to four decimal places.) (a) x is more than 60 (b) x is less than 110 (c) x is between 60 and 110 (d) x is greater than 125 (borderline diabetes starts at 125)

a) .8944 b) .8944 c) .7888 d) .0228

Quick Start Company makes 12-volt car batteries. After many years of product testing, the company knows that the average life of a Quick Start battery is normally distributed, with a mean of 43.2 months and a standard deviation of 6.5 months. (a) If Quick Start guarantees a full refund on any battery that fails within the 36-month period after purchase, what percentage of its batteries will the company expect to replace? (Round your answer to two decimal places.) (b) If Quick Start does not want to make refunds for more than 8% of its batteries under the full-refund guarantee policy, for how long should the company guarantee the batteries (to the nearest month)?

a) 13.57 b) 34

Assuming that the heights of college women are normally distributed with mean 63 inches and standard deviation 3.3 inches, answer the following questions. (Hint: Use the figure below with mean μ and standard deviation σ.) a) What percentage of women are taller than 63 inches? b) What percentage of women are shorter than 63 inches? c) What percentage of women are between 59.7 inches and 66.3 inches? d) What percentage of women are between 56.4 and 69.6 inches?

a) 50% b) 50% c) 68% d) 95%

a) What percentage of the area under the normal curve lies to the left of μ? b) What percentage of the area under the normal curve lies between μ − σ and μ + σ? c) What percentage of the area under the normal curve lies between μ − 3σ and μ + 3σ?

a) 50% b)68% c) 99.7%

The incubation time for a breed of chicks is normally distributed with a mean of 21 days and standard deviation of approximately 3 days. Look at the figure below and answer the following questions. If 1000 eggs are being incubated, how many chicks do we expect will hatch in the following time periods? (Note: In this problem, let us agree to think of a single day or a succession of days as a continuous interval of time. Assume all eggs eventually hatch.) (a) in 15 to 27 days (b) in 18 to 24 days (c) in 21 days or fewer (d) in 12 to 30 days

a) 950 b) 680 c) 500 d) 997

Suppose x has a distribution with μ = 30 and σ = 17. (a) If random samples of size n = 16 are selected, can we say anything about the x distribution of sample means? (b) If the original x distribution is normal, can we say anything about the x distribution of random samples of size 16? c) Find P(26 ≤ x ≤ 31). (Round your answer to four decimal places.)

a) No, the sample size is too small. b) Yes, the x distribution is normal with mean μ x = 30 and σ x = 4.25. c) .4212

Fawns between 1 and 5 months old have a body weight that is approximately normally distributed with mean μ = 27.9 kilograms and standard deviation σ = 3.8 kilograms. Let x be the weight of a fawn in kilograms. a)If a fawn weighs 14 kilograms, would you say it is an unusually small animal? Explain using z values and the figure above. b)If a fawn is unusually large, would you say that the z value for the weight of the fawn will be close to 0, −2, or 3? Explain

a) Yes. This weight is 3.66 standard deviations below the mean; 14 kg is an unusually low weight for a fawn. b) It would have a large positive z, such as 3.

(a) If we have a distribution of x values that is more or less mound-shaped and somewhat symmetric, what is the sample size n needed to claim that the distribution of sample means x from random samples of that size is approximately normal? n ≥ __?__ (b) If the original distribution of x values is known to be normal, do we need to make any restriction about sample size in order to claim that the distribution of sample means x taken from random samples of a given size is normal?

a) n ≥ 30 b) no

Look at the normal curve below, and find μ, μ + σ, and σ.

μ = 28 μ + σ = 30 σ = 2