stats test #3 6.3-6.5

An elevator has a placard stating that the maximum capacity is 2136 lb—12 passengers.​ So, 12 adult male passengers can have a mean weight of up to 2136/12=178 pounds. If the elevator is loaded with 12 adult male​ passengers, find the probability that it is overloaded because they have a mean weight greater than 178 lb.​ (Assume that weights of males are normally distributed with a mean of 180 lb and a standard deviation of 26 lb​.) Does this elevator appear to be​ safe?

.6064 No, there is a good chance that 12 randomly selected adult male passengers 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?

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

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.

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.

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.

The weights of a certain brand of candies are normally distributed with a mean weight of 0.8554 g and a standard deviation of 0.0518 g. A sample of these candies came from a package containing 469 ​candies, and the package label stated that the net weight is 400.4 g.​ (If every package has 469 ​candies, the mean weight of the candies must exceed 400.4469=0.8537 g for the net contents to weigh at least 400.4 ​g.) a. If 1 candy is randomly​ selected, find the probability that it weighs more than 0.8537 g. The probability is b. If 469 candies are randomly​ selected, find the probability that their mean weight is at least 0.8537 g. The probability that a sample of 469 candies will have a mean of 0.8537 g or greater is c. Given these​ results, does it seem that the candy company is providing consumers with the amount claimed on the​ label? (yes, no) because the probability of getting a sample mean of 0.8537 g or greater when 469 candies are selected (is, is not) exceptionally small

a. . 5120 b. .7612 c. yes, is not

The overhead reach distances of adult females are normally distributed with a mean of 205 cm and a standard deviation of 8 cm. a. Find the probability that an individual distance is greater than 217.50 cm. b. Find the probability that the mean for 25 randomly selected distances is greater than 203.50 cm. c. Why can the normal distribution be used in part​ (b), even though the sample size does not exceed​ 30?

a. .0594 b. .8264 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=74.0 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 less than 77 beats per minute. b. If 25 adult females are randomly​ selected, find the probability that they have pulse rates with a mean less than 77 beats per minute. c. Why can the normal distribution be used in part​ (b), even though the sample size does not exceed​ 30?

a. .5948 b. .8849 c. Since the original population has a normal​ distribution, the distribution of sample means is a normal distribution for any sample size.

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

Which of the following statistics are unbiased estimators of population​ parameters?

range, median, variance