Stat - Chapter 8
The distribution of the sample mean, x overbar, will be normally distributed if the sample is obtained from a population that is normally distributed, regardless of the sample size.
True
The standard deviation of the sampling distribution of x overbar, denoted σx, is called the _____ _____ of the _____.
standard error of the mean
Determine μx and σx from the given parameters of the population and sample size. μ = 82, σ = 24, n = 64
μ x overbar = 82 σ x overbar = 3
Suppose a simple random sample of size n is drawn from a large population with mean μ and standard deviation σ. The sampling distribution of x overbar has mean μx=______ and standard deviation σx=______
mean symbol and standard deviation symbol/ root of n
Complete the sentence below. The _____ _____, denoted p with, is given by the formula p =_____, where x is the number of individuals with a specified characteristic in a sample of n individuals.
sample proportion, x/n
Is the statement below true or false? The mean of the sampling distribution of ModifyingAbove p with caretp is p.
The statement is true. The mean of the sample distribution of the sampling proportion equals the population proportion, p.
Suppose the lengths of the pregnancies of a certain animal are approximately normally distributed with mean μ=261 days and standard deviation σ=13 days. Complete parts (a) through (c). (a) What is the probability that a randomly selected pregnancy lasts less than 256 days? (b) What is the probability that a random sample of 16 pregnancies has a mean gestation period of less than 256 days? (c)
Use StatCrunch (a) 0.3503 (b) 0.0620 (c) 0.0058
Suppose a geyser has a mean time between eruptions of 94 minutes. Let the interval of time between the eruptions be normally distributed with standard deviation 37 minutes. Complete parts (a) through (e) below. (a) What is the probability that a randomly selected time interval between eruptions is longer than 112 minutes? (b) What is the probability that a random sample of 9 time intervals between eruptions has a mean longer than 112 minutes? (c) What is the probability that a random sample of 22 time intervals between eruptions has a mean longer than 112 minutes? (d) What effect does increasing the sample size have on the probability? Provide an explanation for this result. Fill in the blanks below. (e) What might you conclude if a random sample of 22 time intervals between eruptions has a mean longer than 112 minutes? Select all that apply.
(a) 0.3133 (b) 0.0722 (c) 0.0112 (d) If the population mean is less than 112 minutes, then the probability that the sample mean of the time between eruptions is greater than 112 minutes decreases because the variability in the sample mean decreases as the sample size increases. (e) The population mean is 94, and this is just a rare sampling.; The population mean may be greater than 94.
Complete parts (a) through (d) for the sampling distribution of the sample mean shown in the accompanying graph. (a) What is the value of μx overbar? (b) What is the value of σx overbar? (c) If the sample size is n=9, what is likely true about the shape of the population?
(a) 300 (b) 30 (c) The shape of the population is approximately normal. (d) 90
Describe the sampling distribution of p with caretp. Assume the size of the population is 25,000. n=400, p=0.1 (a) Choose the phrase that best describes the shape of the sampling distribution of p with caretp below. (b) Determine the mean of the sampling distribution of p with caretp. (c) Determine the standard deviation of the sampling distribution of p with caretp.
(a) Approximately normal because n≤0.05N and np(1−p)≥10. (b) Mean=0.1 (c) Standard Deviation=0.015
According to a survey in a country, 25% of adults do not have any credit cards. Suppose a simple random sample of 400 adults is obtained. (a) Describe the sampling distribution of p with caretp, the sample proportion of adults who do not have a credit card. Choose the phrase that best describes the shape of the sampling distribution of p with caretp below. (b) Determine the mean of the sampling distribution of p with caretp. (c) Determine the standard deviation of the sampling distribution of p with caretp. (d) In a random sample of 400 adults, what is the probability that less than 24% have no credit cards? (e) Would it be unusual if a random sample of 400 adults results in 116 or more having no credit cards?
(a) Approximately normal because n≤0.05N and np(1−p)≥10. (b) Mean=0.25 (c) Standard Deviation=0.022 (d) 0.3247 (e) The result is unusual because the probability that p with caretp is greater than or equal to this sample proportion is less than 5%.
Suppose a simple random sample of size n=200 is obtained from a population whose size is N=20,000 and whose population proportion with a specified characteristic is p=0.6. Complete parts (a) through (c) below. (a) Describe the sampling distribution of p with caretp. (b) Determine the mean of the sampling distribution of p with caretp. (c) Determine the standard deviation of the sampling distribution of p with caretp. (d) What is the probability of obtaining x=126 or more individuals with the characteristic? That is, what is P(p with caretp ≥0.63)? (e) What is the probability of obtaining x=108 or fewer individuals with the characteristic? That is, what is P(p with caretp ≤0.540.54)?
(a) Approximately normal because n≤0.05N and np(1−p)≥10. (b) Mean=0.6 (c) Standard Deviation=0.034641 (d) 0.1932 (e) 0.0416
Suppose a simple random sample of size n=1000 is obtained from a population whose size is N=2,000,000 and whose population proportion with a specified characteristic is p equals 0.44 .p=0.44. Complete parts (a) through (c) below. (a) Describe the sampling distribution of p with caretp. (b) What is the probability of obtaining x=460 or more individuals with the characteristic?
(a) Approximately normal, caretμp=0.440.44 and caretσp almost equals≈0.0157 (b) 0.1014 (c) 0.0280
Consider a random variable X that is normally distributed. Complete parts (a) through (d) below. (a) If a random variable X is normally distributed, what will be the shape of the distribution of the sample mean? (b) If the mean of a random variable X is 30, what will be the mean of the sampling distribution of the sample mean? (c) As the sample size n increases, what happens to the standard error of the mean? (d) If the standard deviation of a random variable X is 10 and a random sample of size n=15 is obtained, what is the standard deviation of the sampling distribution of the sample mean?
(a) Normal (b) 30 (c) The standard error of the mean decreases. (d) 10/square root of 15
(a) Suppose a simple random sample of size n is obtained from a population whose distribution is skewed right. As the sample size n increases, what happens to the shape of the distribution of the sample mean? (b) For the three probability distributions shown, rank each distribution from lowest to highest in terms of the sample size required for the distribution of the sample mean to be approximately normally distributed.
(a) The distribution becomes approximately normal. (b) B, A, C
A simple random sample of size n=36 is obtained from a population with μ=73 and σ=6. (a) Describe the sampling distribution of x overbar. (b) What is P (x overbar >74)? (c) What is Upper P (x overbar ≤70.8)? (d) What is P (71.9<x overbar <75.35)?
(a) The distribution is approximately normal. mean=73 standard deviation=1 Use StatCrunch (b) 0.1587 (c) 0.0139 (c) 0.8549
According to a study conducted by a statistical organization, the proportion of Americans who are satisfied with the way things are going in their lives is 0.82. Suppose that a random sample of 100 Americans is obtained. Complete parts (a) through (c). (a) Describe the sampling distribution of p with caretp. Choose the correct answer below. (b) Using the distribution from part (a), what is the probability that at least 80 Americans in the sample are satisfied with their lives? (c) Using the distribution from part (a), what is the probability that 77 or fewer Americans in the sample are satisfied with their lives?
(a) The distribution of p with caretp can be approximated by a normal distribution with mean=0.82 and standard deviation=0.038. (b) 0.7007 (c) 0.0941
The shape of the distribution of the time required to get an oil change at a 20-minute oil-change facility is unknown. However, records indicate that the mean time is 21.5 minutes, and the standard deviation is 4.3 minutes. Complete parts (a) through (c). (a) To compute probabilities regarding the sample mean using the normal model, what size sample would be required? (b) What is the probability that a random sample of n=40 oil changes results in a sample mean time less than 20 minutes? (c) Suppose the manager agrees to pay each employee a $50 bonus if they meet a certain goal. On a typical Saturday, the oil-change facility will perform 40 oil changes between 10 A.M. and 12 P.M. Treating this as a randomsample, there would be a 10% chance of the mean oil-change time being at or below what value? This will be the goal established by the manager.
(a) The sample size needs to be greater than or equal to 30. (b) The probability is approximately 0.0137. (c) There is a 10% chance of being at or below a mean-oil change time of 20.6 minutes.
True or False: The population proportion and sample proportion always have the same value.
False. The population proportion and sample proportion do not always have the same value.
The reading speed of second grade students is approximately normal, with a mean of 89 words per minute (wpm) and a standard deviation of 10 wpm. Complete parts (a) through (f). (a) What is the probability a randomly selected student will read more than 94 words per minute? (b) What is the probability that a random sample of 10 second grade students results in a mean reading rate of more than 94 words per minute? (c) What is the probability that a random sample of 26 second grade students results in a mean reading rate of more than 96 words per minute? (d) What effect does increasing the sample size have on the probability? Provide an explanation for this result. (e) A teacher instituted a new reading program at school. After 10 weeks in the program, it was found that the mean reading speed of a random sample of 22 second grade students was 93.1 wpm. What might you conclude based on this result? Select the correct choice and fill in the answer box in your choice below. (f) There is a 5% chance that the mean reading speed of a random sample of 20 second grade students will exceed what value?
Use StatCrunch (a) 0.3085 (b) 0.0569 (c) 0.0127 (d) Increasing the sample size decreases the probability because σ overbar x decreases as n increases. (e) A mean reading rate of 94.7 wpm is not unusual since the probability of obtaining a result of 94.7 wpm or more is 0.1285. The new program is not abundantly more effective than the old program. (f) There is a 5% chance of the mean value exceeding 92.8 wpm.