1.)Suppose you interview 10 randomly selected workers and ask how many miles they commute to work. You'll compute the sample mean commute distance. Now imagine repeating the survey many, many times, each time recording a different sample mean commute distance. In the long run, a histogram of these sample means represents:
a) the bias, if any, which is present in the sampling method
b)the true population average commute distance
c)simple random sample
d)the sampling distribution of the sample mean
I chose choice "d" but I was torn between that and choice "b"
2)The average age of residents in a large residential retirement community is 69 years with standard deviation 5.8 years. A simple random sample of 100 residents is to be selected, and the sample mean age of these residents is to be computed.
We know the random variable has approximately a normal distribution because of
Choose one answer.
a. the central limit theorem.
b. the law of large numbers.
c. the 68–95–99.7 rule.
d. the population we're sampling from has a Normal distribution.
I chose choice "a"
the central limit theorem.
1) The correct answer is d) the sampling distribution of the sample mean. This is because in the given scenario, you are repeatedly taking random samples of workers and calculating the mean commute distance for each sample. The histogram of these sample means represents the distribution of the sample means, which is known as the sampling distribution of the sample mean. This distribution shows us the range of possible sample means we could obtain from repeated sampling.
It is understandable why you were torn between choice d) and choice b) (the true population average commute distance). While it is true that the sample means can provide an estimate of the true population average commute distance, in this particular question, we are specifically interested in the distribution of the sample means, rather than the true population parameter itself.
2) The correct answer is a) the central limit theorem. The central limit theorem states that for a large enough sample size, the sample mean (in this case, the average age of residents) will be approximately normally distributed, regardless of the shape of the population distribution. Since we are dealing with a large enough sample (100 residents), we can apply the central limit theorem and conclude that the random variable (the sample mean age) will have an approximately normal distribution.
1) The correct answer is d) the sampling distribution of the sample mean.
In this scenario, you are repeating the survey multiple times and recording different sample mean commute distances each time. The histogram of these sample means represents the sampling distribution of the sample mean. This distribution shows the variability in the sample means that could occur due to random sampling. It gives us information about the range of possible sample means and their likelihood of occurrence. It does not directly represent the true population average commute distance, but it helps us estimate it by providing information about the distribution of sample means.
2) The correct answer is a) the central limit theorem.
The central limit theorem states that the distribution of sample means approaches a normal distribution as the sample size increases, regardless of the shape of the population distribution. In this case, the large retirement community has a population with an average age of 69 years and a standard deviation of 5.8 years. When a simple random sample of 100 residents is selected, the sample mean age will tend to follow a normal distribution due to the central limit theorem. This allows us to make inferences and calculations based on the assumption of a normal distribution for the sample mean age.