Statistics 350 ? Homework #5 Solutions 1. Let X = the salary (in $1000s) for a randomly selected full-time worker in a certain city. Suppose the expected (or mean) salary is $62.0 and the standard deviation is $12.0. a. Consider the procedure of selecting a random sample of 64 salaries and computing the sample mean salary, x . Suppose you repeated that procedure many, many times. i. Describe the distribution of the resulting x values, including all relevant features. (Approximate) Normal distribution with mean of $62 and standard deviation of ? $12.0 $1.5 n 64 . X ~ N(62,1.5) ii. Draw this distribution. Be sure to include all appropriate labels and provide the values along the axis that represent 1, 2, and 3 standard deviations from the mean. b. If you were to obtain a random sample of 64 salaries, what is the probability that the sample mean salary will exceed $65? P(X > $65) = P(Z > $65-$62 $1.50 ) = P(Z > 2) = 1 ? P(Z ? 2) = 1 ? 0.9772 = 0.0228 (Note: the empirical rule would be fine here too ? and give 0.025). c. Could you apply the same technique with a random sample of 9 salaries? Explain using one brief sentence. No, as we are not given the model or distribution for salaries for this population, thus we could not apply the same technique because we do not have a large enough sample size (e.g. n > 30) to rely on the Central Limit Theorem. Normal distribution with a mean of $62.0 (thousands) and a standard deviation of $1.5 (thousands). __________________________________________________________________________________________ 2. In a geology course, students were learning to use a balance scale to make accurate weighings of rock samples. One student plans to weigh a rock 20 times and then calculate the average of the 20 measurements to estimate her rock's true weight. A second student plans to weigh a rock 5 times and calculate the average of the 5 measurements to estimate his rock's true weight. Which student is more likely to come to the closest to the true weight of the rock he or she is weighing? Circle one: a. The student who weighed the rock 20 times. b. The student who weighed the rock 5 times. c. Both averages would be equally close to the true weight. 3. Exercise 9.60 on page 390. a. Mean of sampling distribution of x : x = _60 mph_____________________________ Standard deviation of sampling distribution of x : x = _1 mph____________________ ? 6 mp h 1 mp hn 36 b. Fill in the blanks: For a random sample of n = 36 vehicles, there is about a 95% chance that mean vehicle speed in the sample will be between __58__ and __62__ mph. Since we have an approximate normal distribution, 95% of the mean vehicle speeds will be within 2 standard deviations of the mean by the Empirical Rule. 6 0 m p h 2 * s . d . (x ) 6 0 m p h 2 (1 m p h ) c. Using the answer in (b), is a sample mean of 66 mph consistent with the belief that the mean speed at this location is = 60 mph? Yes No Explain in one simple sentence: A sample mean of 66 mph is beyond the range of possible sample means for 95% of all random samples of sample size n=36. _______________________________________________________________________________________________ 4. Consider the distribution of the number of hours that college students spend sleeping on a typical weeknight. This distribution is very skewed to the right, with a mean of 5 and a standard deviation of 1. A researcher plans to take a simple random sample of 18 college students. If we were to imagine that we could take all possible random samples of size 18 from the population of college students, the sampling distribution of average number of hours spent sleeping will have a shape that is (circle one)? a. Exactly normal. b. Less skewed than the population. c. Just like the population (i.e., very skewed to the right). d. It's impossible to predict the shape of the sampling distribution. Kam Hamidieh Homework 9 ? Learning about two population proportions