Exam 12: Asking and Answering Questions About a Population Mean

The width of the one-sample confidence interval for μ decreases as the sample size grows larger.

Under what conditions is it reasonable to use the z confidence interval to estimate a population mean?

As n grows larger, the mean of the sampling distribution of gets closer to μ.

For a random sample from a skewed population, for which of the following sample sizes would the sampling distribution of be closest to normal in shape? ​

Suppose you take a random sample from a population known to be normally distributed, but the value of σ is unknown. Your sample size is n =15. Which formula below should be used to find the 95% confidence interval for the mean? ​

The lecturer decided to estimate the mean mental reading speed for the university students. Randomly, seventy nine students were selected and the speed of their reading was recorded. For this sample, the mean reading speed was 235 words per minute. Suppose that the sample standard deviation was 46 words per minute. Find the margin of error and construct a 95% confidence interval for the mean reading speed for the university students.

The t confidence interval formula for estimating μ should only be used when the population being sampled is at least approximately normally distributed.

What is the standard deviation of for random samples of size 80 from a population with a mean of 48 and a standard deviation of 11? ​

Researchers studied the average height of 12 years old girls in the city. They determined that the mean growth for a representative sample of 10 girls was 59.4 inches. The original sample data values are: ​ Select the most appropriate 90% bootstrap confidence interval for a population mean height of 12 years old girls in the city and its correct interpretation.

Commute distances for employees of a large company have a distribution that is quite skewed. Suppose the mean commute distance is 20 miles and that the standard deviation is 7 miles. What can be said about the mean commute distance for a random sample of size 49 employees from this plant? ​

Suppose that a study has been published linking arsenic, which is commonly found in tap water, to increased cancer rates. The study also stated that arsenic levels below 10 ppm could be considered harmless. Suppose that residents of Hacienda Heights, CA, are worried about the arsenic level in the local tap water. a)What hypothesis should they test? b)Describe a Type I and Type II error in this context.

Under what conditions is it reasonable to use the t confidence interval to estimate a population mean?

The distribution of will always have the same shape as the distribution of the population being sampled.

According to internal testing done by the HardCore Apple Company, the mean firmness of their apples, as measured on the Mohs hardness scale is 7.0, with a standard deviation of 0.3. a)If the claim by Hardcore is true, what is the mean of the sampling distribution of for samples of size n = 16? b)If the claim by HardCore is true, what is the standard deviation of the sampling distribution of for samples of size n = 16? c)If the distribution of HardCore firmness is approximately normal, and their claim about the mean firmness is true, what is the probability that the mean firmness of a random sample of size n = 16 will be less than 6.9?

In an analysis of hunting by African lions, biologists filmed prey captures from the safety of their vehicles. The capture of prey was divided into a sequence of events for study, one of which is the stalk, defined as the reduction of predator-prey distance for prey that has been specifically located and the prey is unaware of or minimally alarmed by the predator. The investigators identified two types of stalk: (a) "crouching,"--the lion is concealed and either the lion advances toward the prey or the prey advances (unaware) toward the lion, and (b) "running,"--the lion is less concealed and advances toward the prey in a rapid manner. Data on lions' stalks of Thomson's and Grant's gazelles from a sample of 151 kills appear in the table below. ​ a)On the basis of monitoring radio-collared lions for many years, biologists believe that the average stalking time for all prey is approximately 25.6 minutes. Do the data above provide evidence that for this population of lions the average time to stalk Thomson's and Grant's gazelles is different from what was originally thought? b)The same monitoring of radio-collared lions over the years has suggested that the overall proportion of stalks that are the crouching type is about 0.87. Do the data above provide evidence that for this population of lions the proportion of crouching stalks of Thomson's and Grant's gazelles is less than what was originally thought?

How are the quantities, and , related?

The department of natural resources defines a fish to be unsafe to eat if the polychlorinated biphenol (PCB) concentration exceeds 5 parts per billion (ppb). A random sample of 10 fish taken from a local lake resulted in the data listed below: 2.6, 6.6, 4.8, 5.4, 5.1, 4.5, 6.9, 4.9, 3.7, 3.8 Is there sufficient evidence to conclude that the mean PCB concentration for fish from this lake exceeds 5 ppb?

Psychologists have noted that people tend make classifications based on their "typicality." For example, a dog is more readily classified as a mammal than is a whale, because dogs are "more typical" of mammals than are whales. Previous large-scale studies have established the following for songbirds such as robins: when young children are shown a picture of a songbird and asked if it is a bird they respond "yes" in an average of 750 milliseconds. Do children regard chickens as representative of birds? If so, are they more representative or less representative of birds than are songbirds? Let μ represent the mean time it takes children to respond to a question about whether a chicken is a bird. The investigators wish to determine whether the classification time differs from the time needed to classify songbirds. a)What is the appropriate null hypothesis? b)What is the appropriate alternative hypothesis? c)In your own words, distinguish between a Type I and a Type II error in this context.

The foraging capability of Downy Woodpeckers can be compromised if part of their habitat disappears due to significant human encroachment. These creatures eat insects in trees, and if they cannot find enough food due to a decreasing forest area their body mass may decrease. A 20-year-old study of Downy Woodpeckers indicates that for male Downy Woodpeckers the average mass is about 28g, with a standard deviation of approximately 1.5g. Suppose it is desired to estimate μ, the average mass of male Downy Woodpeckers in a local forest preserve. If the 20-year-old values are taken as reasonable estimates, determine the sample size necessary to estimate μ to within 0.2 grams with 95% confidence.

Many national parks, wildlife refuges, and wilderness areas lie directly below designated military air space. This raises issues of whether wildlife and, for example, loud jet noises are compatible. In an effort to evaluate the effects of low-altitude jet-noise, researchers simulated these noises for 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, 65.5, 68.4, 64.9, 63.1, 55.3, 63.3 Assume that it is reasonable to think that these deer can be regarded as a random sample of desert mule deer. a)Calculate and interpret a 95% confidence interval for the mean heart rate 3 minutes after loud jet noises. b)The mean heart rate for desert mule deer is about 51.2 beats per minute. Based on the confidence interval in part (a), do you think that the loud jet noise has some lingering effect on the heart rate of desert mule deer? Explain.