Section 6.4 Homework

The standard deviation of the distribution of sample means is ​ _______.

σ/√n

Which of the following is not a commonly used​ practice?

If the distribution of the sample means is normally​ distributed, and ngreater than>​30, then the population distribution is normally distributed.

The​ _______ tells us that for a population with any​ distribution, the distribution of the sample means approaches a normal distribution as the sample size increases.

Central Limit Theorem

Weights of golden retriever dogs are normally distributed. Samples of weights of golden retriever​ dogs, each of size n=​15, are randomly collected and the sample means are found. Is it correct to conclude that the sample means cannot be treated as being from a normal distribution because the sample size is too​ small? Explain.

No; the original population is normally​ distributed, so the sample means will be normally distributed for any sample size.

Which of the following is NOT a conclusion of the Central Limit​ Theorem?

The distribution of the sample data will approach a normal distribution as the sample size increases.

Before every​ flight, the pilot must verify that the total weight of the load is less than the maximum allowable load for the aircraft. The aircraft can carry 41 ​passengers, and a flight has fuel and baggage that allows for a total passenger load of 6,765 lb. The pilot sees that the plane is full and all passengers are men. The aircraft will be overloaded if the mean weight of the passengers is greater than 1656,765 lb/ 41=165 lb. What is the probability that the aircraft is​ overloaded? Should the pilot take any action to correct for an overloaded​ aircraft? Assume that weights of men are normally distributed with a mean of 174 lb and a standard deviation of 37.5.

The probability is approximately 0.9382. Yes. Because the probability is​ high, the pilot should take action by somehow reducing the weight of the aircraft.

An elevator has a placard stating that the maximum capacity is 1630 lb —10 passengers.​ So, 10 adult male passengers can have a mean weight of up to 1630/10=163 pounds. If the elevator is loaded with 10 adult male​ passengers, find the probability that it is overloaded because they have a mean weight greater than 163 lb.​ (Assume that weights of males are normally distributed with a mean of 171 lb and a standard deviation of 25 lb​.) Does this elevator appear to be​ safe?

The probability the elevator is overloaded is 0.8438. ​No, there is a good chance that 10 randomly selected people will exceed the elevator capacity.

A researcher collects a simple random sample of​ grade-point averages of statistics​ students, and she calculates the mean of this sample. Under what conditions can that sample mean be treated as a value from a population having a normal​ distribution?

The sample has more than 30​ grade-point averages. If the population of​ grade-point averages has a normal distribution.

The overhead reach distances of adult females are normally distributed with a mean of 197.5 cm and a standard deviation of 8 cm. a. Find the probability that an individual distance is greater than 210.00 cm. b. Find the probability that the mean for 20 randomly selected distances is greater than 195.30 cm. c. Why can the normal distribution be used in part​ (b), even though the sample size does not exceed​ 30?

a. 0.0594 b. 0.8907 c. The normal distribution can be used because the original population has a normal distribution.

Assume that females have pulse rates that are normally distributed with a mean of μ=75.0 beats per minute and a standard deviation of σ=12.5 beats per minute. Complete parts​ (a) through​ (c) below. a. If 1 adult female is randomly​ selected, find the probability that her pulse rate is between 71 beats per minute and 79 beats per minute. b. If 4 adult females are randomly​ selected, find the probability that they have pulse rates with a mean between 71 beats per minute and 79 beats per minute. c. Why can the normal distribution be used in part​ (b), even though the sample size does not exceed​ 30?

a. 0.251 b. 0.4778 c. Why can the normal distribution be used in part​ (b), even though the sample size does not exceed​ 30?

An engineer is going to redesign an ejection seat for an airplane. The seat was designed for pilots weighing between 150 lb and 191 lb. The new population of pilots has normally distributed weights with a mean of 159 lb and a standard deviation of 25.1 lb. a. If a pilot is randomly​ selected, find the probability that his weight is between 150 lb and 191 lb. b. If 38 different pilots are randomly​ selected, find the probability that their mean weight is between 150 lb and 191 lb. c. When redesigning the ejection​ seat, which probability is more​ relevant?

a. 0.5386 b. 0.9863 c. Part​ (a) because the seat performance for a single pilot is more important.

Assume that females have pulse rates that are normally distributed with a mean of μ=75.0beats per minute and a standard deviation of σ=12.5 beats per minute. Complete parts​ (a) through​ (c) below. a. If 1 adult female is randomly​ selected, find the probability that her pulse rate is less than 78 beats per minute. b. If 25 adult females are randomly​ selected, find the probability that they have pulse rates with a mean less than 78 beats per minute. c. Why can the normal distribution be used in part​ (b), even though the sample size does not exceed​ 30?

a. 0.5948 b. 0.8849 c. Since the original population has a normal​ distribution, the distribution of sample means is a normal distribution for any sample size.

Annual incomes are known to have a distribution that is skewed to the right instead of being normally distributed. Assume that we collect a large n>​30 random sample of annual incomes. Can the distribution of incomes in that sample be approximated by a normal distribution because the sample is​ large? Why or why​ not?

​No; the sample means will be normally​ distributed, but the sample of incomes will be skewed to the right.