Intro to Statistics Chapter 7.2 Sampling Distribution

a.) Since Sigma is almost as large as mu and 0 is a lower bound only 1.33 standard deviations below the mean, the distribution is right-skewed. b.) With a random sample, the data distribution picks up the characteristics from the population distribution, so we anticipate it as right skewed. c.) With n=45, sampling distribution of sample mean is bell shaped b the central limit theorem.

7.21 Shared family phone plan A recent personalized information sheet from your wireless phone carrier claims that the mean duration of all your phone calls was mu= 2.8 minutes with a standard deviation of Sigma = 2.1 minutes. a.) Is the population distribution of the duration of your phone calls likely to be bell-shaped, right r left-skewed? b.) You are on a shared wireless plan with your parents, who are statisticians. They look at some of your recent monthly statements that list each call and its duration and randomly sample 45 calls from the thousands listed there. They construct a histogram of the duration to look at the data distribution. Is this distribution likely to be bell-shaped right or left-skewed?

a.) Mean = 8.20 Standard deviation = 0.30 b.) 0.994

7.23 Restaurant profit? Jan's All You Can Eat Restaurant charges $8.95 per customer to eat at the restaurant. Restaurant management finds that its expense per customer based on how much the customer eats and the expense of labor has a distribution that is skewed to the right with a mean of $8.20 and a standard deviation of $3. a.) If the 100 customers on a particular day have the characteristics of a random sample from their customer base, find the mean and standard deviation of the sampling distribution of the restaurant's sample mean expense per customer. b.) Find the probability that the restaurant makes a profit that day, with the sample mean expense being less than $8.95. (Hint: Apply the central limit theorem to the sampling distribution in part a.).

a.) Mean = 130 Standard deviation = 3.46 b.) Normal. If the population distribution is approximately normal, then the sampling distribution is approximately normal for any same size. c.) 0.002

7.25 Blood pressure Vincenzo Baranello was diagnosed with high blood pressure. He was able to keep his blood pressure under control for several months by taking blood pressure medicine (amlodipine besylate). Baranello's blood pressure is monitored by taking three reading a day, in the early morning, at midday, and in the evening. a.) During this period, the probability distribution of his systolic blood pressure reading had a mean of 130 and a standard deviation of 6. If the successive observations behave like a random sample from this distribution, find the mean and standard deviation of the sampling distribution of the sample mean for the three observations each day. b.) Suppose that the probability distribution of his blood pressure reading is normal. What is the shape of the sampling distribution? Why? c.) Refer to part b. Find the probability that the sample mean exceeds 140, which is considered problematically high. (Hint: Use the sampling distribution, not the probability distribution for each observation.)

mu

Symbol for population distribution

p

Symbol for population proportion (mean)

sigma

Symbol for standard deviation of population distribution

a.) Mean is $74,550, and the standard deviation is $19,872; likely skewed right because of a few large salaries. b.) $75,207 and $18,901; shape similar to population distribution. c.) $74,550 and $1987; approximately normal distribution due to Central Limit Theorem. d.) $100,000 is barely one standard deviation above the population mean, ut it is more than 12 standard deviations above the mean of the sampling distribution of the sample mean.

7.27 Average monthly sales A large corporation employes 27,251 individuals. The average income in 2008 for all employees was $74,550 with a standard deviation of $19,872. You are interested in comparing the incomes of today's employees with those of 2008. A random sample of 100 employees of the corporation yields x bar = $75207 and s= $18,901. a.) Describe the center and variability of the population distribution. What shape does it probably have? Explain. b.) Describe the center and variability of the data distribution. What shape does it probably have? Explain. c.) Describe the center and variability of the sampling distribution of the sample mean for n=100. What shape does it have? Explain. d.) Explain why it would not be unusual to observe an individual who earns more than $100,000, but it would be highly unusual to observe a sample mean income of more than $100,000 for a random sample size of 100 people.

a.) Mean = 0.70 Standard deviation = 0.065 b.) Bell-shaped by the central limit theorem c.) Probability = 0.06

7.31

a.) np = 200(1/9)= 22.2 n(1-p)=200(8/9)= 177.8 Number of successes and number of failures are both larger than 15 Shape is approximately normal with mean = p = 1/9 and standard deviation = sqrt(p(1-p)/n) = sqrt((1/9) (8/9)/200 = 0.022 b.) Shape is approximately normal with number of successes and number of failures are both larger than 15, with mean = p = 1/9 and standard deviation = sqrt ((1/9)(8/9)/800) = 0.011

7.33

a.) Bell shape with mean = $84,396 Standard deviation= $2197.50. b.) z= -2.00 c.) 0.93

7.39

Sampling distribution

The probability distribution of a sample statistic, such as a sample proportion or sample mean. With random sampling, it provides probabilities for all the possible values of the statistic.