Exam 9: Sampling Distributions

If the population distribution is skewed, in most cases the sampling distribution of the sample mean can be approximated by the normal distribution if the samples contain at least 30 observations.

Given that X is a binomial random variable with very large n, the binomial probability P(X $\le$ 5) is approximated by the area under a normal curve to the left of

If two random samples of sizes n₁ and n₂ are selected independently from two non-normally distributed populations, then the sampling distribution of the sample mean difference, , is

DVD Rental Store A DVD rental store wants to know what proportion of its customers are under age 21.A simple random sample of 500 customers was taken, and 375 of them were under age 21.Presume that the true population proportion of customers under age 21 is 0.68. -{DVD Rental Store Narrative} What is the probability that the sample proportion is within 0.03 of the true proportion of customers who are under age 21?

In general, the binomial probability P(X $\ge$ x) is approximated by the area under the normal curve to the left of x - .5.

CFO's Salary Suppose that the starting salaries of female CFO's have a positively skewed distribution with mean of $56,000 and a standard deviation of $12,000.The starting salaries of male CFO's are positively skewed with a mean of $50,000 and a standard deviation of $10,000.A random sample of 50 female CFO's and a random sample of 40 male CFO's are selected. -{CFO's Salary Narrative} Find the expected value of the sample mean difference.

As a general rule in computing the standard error of the sample mean, the finite population correction factor is used only if the:

The amount of time it takes to complete a final examination is negatively skewed distribution with a mean of 70 minutes and a standard deviation of 8 minutes.If 64 students were randomly sampled, the probability that the sample mean of the sampled students exceeds 73.5 minutes is approximately 0.

In a given year, the average annual salary of a NHL hockey player was $205,000 with a standard deviation of $24,500.If a simple random sample of 50 players was taken, what is the probability that the sample mean will exceed $210,000?

Average Annual Revenue Suppose that the average annual revenue of a small business is $150,000 with a standard deviation of $40,000.Assume that the revenue distribution is normal. -{Average Annual Revenue Narrative} What is the probability that the average annual revenue of a random sample of 4 businesses is less than $120,000?

The amount of bleach a machine pours into bottles has a mean of 50 ounces with a standard deviation of 0.25 ounces.Suppose we take a random sample of 36 bottles filled by this machine.The sampling distribution of the sample mean has a standard error of 0.25 ounces.

If the population is normal, then is normally distributed for __________________ values of n.

Number of Fish Tanks The following data give the number of fish tanks owned for a population of 4 families. -{Number of Fish Tanks Narrative} List all possible samples of 2 families that can be selected without replacement from this population, and compute the sample mean for each sample.

If two random samples of sizes 30 and 36 are selected independently from two populations with means 78 and 85, and standard deviations 12 and 15, respectively, then the mean of the difference is equal to:

Barack Obama Bill Clinton, the former President of the United States, believes that the proportion of voters who will vote for Barack Obama in the year 2012 presidential elections is 0.65.A sample of 500 voters is selected at random. -{Barack Obama Narrative} Find the expected value and the standard deviation of the sample proportion .

In general, the binomial probability P(X = x) is approximated by the area under a normal curve between x - .5 and x + .5.

What is the probability that you get more than 200 heads if you flip a coin 400 times?

Number of Fish Tanks The following data give the number of fish tanks owned for a population of 4 families. -{Number of Fish Tanks Narrative} A sample of size 2 is drawn at random from the population.Use the formulas and to calculate the mean and the standard deviation of the sampling distribution of the sample means.

Given a binomial random variable with n = 15 and p = .40, find the exact probabilities of the following events and their normal approximations. a. X = 6 b. X $\ge$ 9 c. X $\le$ 10

As a general rule, the normal distribution is used to approximate the sampling distribution of the sample proportion only if the sample size n is greater than or equal to 30.