Exam 12: Asking and Answering Questions About a Population Mean

All other things being equal, choosing a smaller value of α will increase the probability of making a type II error.

The State Fisheries Department wishes to stock the Styx River with fish, and would like the species to not only survive but thrive. The "substrate" (pebble size at bottom) of a river is an important determinant of the quality of spawning habitat. Unknown to the Fisheries officials, the pebble diameters in the Styx River are approximately normally distributed with a mean of 24 mm, and a standard deviation of 8 mm. Fisheries officials will select a random sample of pebbles in an attempt to estimate the average pebble size. a) On the scale below, sketch two curves, one representing the distribution of the original population and the other the sampling distribution of $\bar { x }$ for a sample of size $n = 16$ . Be sure to indicate which curve is which. What are the mean and standard deviation of the sampling distribution of $\bar { x }$ ? b) What is the approximate probability that a sample of n = 16 from this population would result in a sample mean greater than 30 mm? The graph below is a relative frequency histogram of the lengths of the gentle Hecate fish in the Styx River, as recorded during a large fish tagging survey. You can assume that this histogram represents the population of fish lengths. c) Suppose you were to take a random sample, n = 4, from this population of fish and measure their lengths. Describe the shape, center, and spread of the sampling distribution of x , as compared to the population distribution.

Local health authorities are concerned that the hectic pace in their city has elevated the stress, and thus the blood pressure, of women. The authorities will study this issue by measuring the blood pressure of a random sample of women in the city and estimating the mean blood pressure, μ , of women in the city. Blood pressure is approximately normally distributed in humans, and a previous study suggests the population standard deviation of women's blood pressure is about 8 mm Hg. Suppose the health authorities accept this value as a reasonable estimate of the standard deviation of blood pressure of today's women in this city. If it is desired to estimate μ to within 1.0 mm Hg with 95% confidence, what is the minimum sample size that is necessary?

Describe in a few sentences how each of the following affects the power of a test: a) The size of the difference between the actual value and the hypothesized value of the population mean. b) The significance level, $\alpha$ c) The sample size

Logging activity in forests is thought to affect the behavior of black bears (Ursus americanus). An important measure of animal behavior is the home range, the area used by animals in their lives. In a study of black bears in a logged Canadian forest, the spring and early summer home range (in $\mathrm { km } ^ { 2 }$ ) of 12 radio-collared female black bears was measured with the following results: $39.9,23.5,42.1,29.4,34.4,40.9,27.9,22.3,13.0,20.1,13.3,8.6$ a) Construct and interpret a $95 \%$ confidence interval for the mean home range of female black bears in this logged forest. b) The typical home range of females in forests with no logging is $20 \mathrm {~km} ^ { 2 }$ . Based on the confidence interval from part (a), do you think that the mean home range size of females in this logged forest could be the same as the mean home range size in non-logged forests? Explain, using appropriate statistical terminology.

One indicator of the health of field mice is the length of their tails. The following data on tail length (in mm) were gathered from a representative sample of field mice near Fairfield, Iowa. A biologist would like to construct a 95% confidence interval for the population mean tail length of the field mice near Fairfield. Tail lengths of field mice () 56.1 59.1 59.2 59.3 59.5 59.7 59.7 59.8 60.0 60.1 60.2 60.3 60.3 60.3 60.4 60.4 60.5 60.5 60.6 60.6 60.7 60.8 60.9 61.4 a) Using a graphical display of your choice, display the data in a way that will allow you to determine whether it would be appropriate to use a t confidence interval to estimate the population mean. b) Using the graphical display from part (a), would you advise that constructing a t confidence interval is appropriate? Explain the basis of your recommendation.

Describe in a few sentences how each of the following affects the power of a test: a) The size of the difference between the actual value and the hypothesized value of the population mean. b) The significance level, $\alpha$ c) The sample size

A A random sample of size $n = 36$ is selected from a population with mean $\mu = 50$ and standard deviation $\sigma = 12$ . Determine the mean and standard deviation of the sampling distribution of $\bar { x }$ .

Give as much information as you can about the P-value of a t-test in the following situations: a) Two-tailed test, $n = 23 , t = 2.1$ b) Lower-tailed test, $n = 12 , t = - 1.6$

The level of significance of a test is the probability of making a type I error.

Give as much information as you can about the P-value of a t-test in the following situations: a) Two-tailed test, $n = 14 , t = 2.4$ b) Upper-tailed test, $n = 22 , t = 1.9$

When the hatching of young geese is very near, the father guards the nest to defend it from predators that may be attracted by the hatchlings noisy entrance into the world. The following data are the typical distances from the nest for 24 soon to be father geese. A biologist would like to construct a $95 \%$ confidence interval for the mean distance of future father geese from the nest during this period. (Distances are in $\mathrm { km }$ .) 2.0 4.4 4.5 4.5 4.7 4.9 4.9 5.0 5.2 5.3 5.3 5.4 5.4 5.4 5.5 5.5 5.6 5.6 5.6 5.6 5.7 5.8 5.9 6.3 a) Using a graphical display of your choice, display the data in a way that will allow you to determine whether it would be appropriate to use a $95 \% t$ confidence interval to estimate the population mean. b) Using the graphical display from part (a), would you advise that constructing a $95 \%$ confidence interval is appropriate? Explain the basis of your recommendation.

A very large study of college students' study habits found that the time (in hours) that freshmen study each week is approximately normal with mean 24 hours and standard deviation 8 hours. Consider random samples of size 16 from the population of freshmen. a) On the axes below, sketch curves representing the distribution of the original population and the sampling distribution of $\bar { x }$ for samples of size $n = 16$ . Be sure to indicate which curve is which. What are the mean and standard deviation of the sampling distribution of $\bar { x }$ ? b) What is the probability that a sample of size $n = 16$ from this population would result in a sample mean greater than 30 hours? The study of college freshmen also asked about the students' time spent playing video games. Suppose that the population relative frequency histogram for time spend playing video games is as shown below. The mean amount of time spent playing video games was $0.8$ hours per week. c) Suppose you were to take a random sample, $n = 4$ , from this population of freshmen and ask them how much time they spent playing video games per week. Describe the shape, center, and spread of the sampling distribution of $\bar { x }$ , as compared to the population distribution.

Many national parks, wildlife refuges, and wilderness areas lie directly below designated military air space. This raises issues of whether wildlife and loud jet noises are compatible. In an effort to evaluate the effects of low-altitude jet-noise, researchers simulated these noises for a representative sample of captive desert mule deer. Three minutes after the loud noises were played, the heart rates (beats per minute) of these animals was measured, with the following results: $55.9,51.1,57.5 , \quad 65.5,68.4,64.9,63.1,55.3,63.3$ a) Calculate and interpret a 95% confidence interval for the mean heart rate 3 minutes after loud jet noises. b) The usual mean heart rate for desert mule deer is about 51.2 beats per minute. Does the confidence interval in part (a) support the view that the loud jet noise has some lingering effect on the heart rate of desert mule deer? Explain, using appropriate statistical language.

When performing hypothesis tests, there are assumptions that must be met in order for the test to be appropriate. Describe the assumptions associated with a hypothesis test about a population mean when the sample size is small. For any assumptions that can be checked using sample data, describe how the check would be performed.