Chapter 6-4

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.

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

An elevator has a placard stating that the maximum capacity is 1580 lb - 10 passengers. So, 10 adult male passengers can have a mean weight of up to 1580/10=158 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 158 lb. (Assume that weights of males are normally distributed with a mean of 164 lb and a standard deviation of 34 lbs.) Does the elevator appear to be safe? The probability the elevator is overloaded is _____. Does the elevator appear to be safe? a) No, there is a good change that 10 randomly selected people will exceed the elevator capacity. b) Yes, 10 randomly selected people will always be under the weight limit. c) Yes, there is a good change that 10 randomly selected people will not exceed the elevator capacity. d) No, 10 randomly selected people will never be under the weight limit.

The probability the elevator is overloaded is _0.7116_. 34/sq rt. 10=10.75 normalcdf(158,10000,164,10.75)=0.7116 Does the elevator appear to be safe? a) No, there is a good change that 10 randomly selected people will exceed the elevator capacity.

The standard deviation of the distribution of sample means is __________.

The standard deviation of the distribution of sample means is _o/sq. rt. n_.

A ski gondola carries skiers to the top of a mountain. Assume that weights of skiers are normally distributed with a mean of 182 lbs. and a standard deviation of 45 lbs. The gondola has a stated capacity of 25 passengers, and the gondola is rated for a load limit of 3500 lb. Complete parts (a) through (d) below. a) Given that the gondola is rated for a load limit of 3500 lbs., what is the maximum mean weight of the passengers if the gondola is filled to the stated capacity of 25 passengers? _____ lb. b) If the gondola is filled with 25 randomly selected skiers, what is the probability that their mean weight exceeds the value from part (a)? _____ c) If the weight assumptions were revised so that the new capacity became 20 passengers and the gondola is filled with 20 randomly selected skiers, what is the probability that their mean weight exceeds 175 lbs., which is the maximum mean weight that does not cause the total load to exceed 3500 lbs? _____ d) Is the new capacity of 20 passengers safe? Since the probability of overloading is __________ the new capacity __________ to be safe enough.

a) Given that the gondola is rated for a load limit of 3500 lbs., what is the maximum mean weight of the passengers if the gondola is filled to the stated capacity of 25 passengers? _140_ lb. 3500/25=140 b) If the gondola is filled with 25 randomly selected skiers, what is the probability that their mean weight exceeds the value from part (a)? _1_ 45/sq. rt. 25=9 normalcdf(-10000,3500,140,9)=1 c) If the weight assumptions were revised so that the new capacity became 20 passengers and the gondola is filled with 20 randomly selected skiers, what is the probability that their mean weight exceeds 175 lbs., which is the maximum mean weight that does not cause the total load to exceed 3500 lbs? _0.9366_ d) Is the new capacity of 20 passengers safe? Since the probability of overloading is _over 50%,_ the new capacity _does not appear_ to be safe enough.

Assume that females have pulse rates that are normally distributed with a mean of u=73.0 beats per minute and a standard deviation of o=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 66 beats per minute and 80 beats per minute. _____ b) If 25 adult females are randomly selected, find the probability that they will have pulse rates with a mean between 66 beats per minute and 80 beats per minute. _____ c) Why can the normal distribution be used in part (b), even though the sample size does not exceed 30? A) Since the distribution is of individuals, not sample means, the distribution is a normal distribution for any sample size. B) Since the original population has a normal distribution, the distribution of sample means is a normal distribution for any sample size. C) Since the distribution is of sample means, not inidividuals, the distribution is a normal distribution for any sample size. D) Since the mean pulse rate exceeds 30, the distribution of sample means is a normal distribution for any sample size.

a) If 1 adult female is randomly selected, find the probability that her pulse rate is between 66 beats per minute and 80 beats per minute. _0.4245_ normalcdf(66,80,73,12.5)=0.4245 b) If 25 adult females are randomly selected, find the probability that they will have pulse rates with a mean between 66 beats per minute and 80 beats per minute. _0.9949_ o/sq. rt. 25=12.5/sq. rt. 25=2.5 normalcdf(66,80,73,2.5)=0.9949 c) Why can the normal distribution be used in part (b), even though the sample size does not exceed 30? B) Since the original population has a normal distribution, the distribution of sample means is a normal distribution for any sample size.

Assume that females have pulse rates that are normally distributed with a mean of u=72.0 beats per minute and a standard deviation of o=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 76 beats per minute. _____ b) If 16 adult females are randomly selected, find the probability that they have pulse rates with a mean less than 76 beats per minute. _____ c) Why can the normal distribution be used in part (b), even though the sample size does not exceed 30? A) Since the distribution is of individuals, not sample means, the distribution is a normal distribution for any sample size. B) Since the mean pulse rate exceeds 30, the distribution of sample means is a normal distribution for any sample size. C) Since the distribution is of sample means, not individuals, the distribution is a normal distribution for any sample size. D) Since the original population has a normal distribtuion, the distribution of sample means is a normal distribution for any sample size.

a) If 1 adult female is randomly selected, find the probability that her pulse rate is less than 76 beats per minute. _0.6255_ normalcdf(-10000,76172,12.5)=0.6255 b) If 16 adult females are randomly selected, find the probability that they have pulse rates with a mean less than 76 beats per minute. _0.8997_ o/sq. rt. of 16= 12.5/sq. rt. 16=3.125 normalcdf(-10000,76,72,3.125)=0.8997 c) Why can the normal distribution be used in part (b), even though the sample size does not exceed 30? D) Since the original population has a normal distribtuion, the distribution of sample means is a normal distribution for any sample size.

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? a) If the population of statistics has a normal distribution. b) If the population of grade-point averages has a normal distribution. c) The researcher collects more than 30 samples. d) The sample has more than 30 grade-point averages.

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

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? a) Yes; the sample size is over 30, so the sample of incomes will be normally distributed. b) No; the sample means will be normally distributed, but the sample of incomes will be skewed to the right. c) No; the population of incomes is not normally distributed, so the sample means will not be normally distributed for any sample size. d) No; unless more than 30 samples are collected, the sample of incomes wil not be normally distributed.

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

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. a) Yes; the sample size must be over 30 for the sample means to be normally distributed. b) No; the samples are collected randomly, so the sample means will be normally distributed for any sample size. c) No; as long as more than 30 samples are collected, the sample means will be normally distributed. d) No; the original population is normally distributed, so the sample means will be normally distributed for any sample size.

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