(Solved): 1. The value of the population parameter varies from sample to sample. 2. The sampling distribution ...

1. The value of the population parameter varies from sample to sample. 2. The sampling distribution of sample means is a probability distribution of sample means with a mean equal to the population mean and a standard deviation equal to the standard error. 3. The sampling distribution of sample means of all samples of size 8 is a perfectly normal distribution. 4. A \( 95 \% \) confidence interval refers to our confidence in the process used to estimate the sample mean. Suppose that the endpoints of a \( 90 \% \) confidence interval for a population mean are 10 and \( 14.8 \). Specify if the following statements are True or False. 5. There is a \( 50 \% \) chance that the population mean is not between 10 and \( 14.8 \) 6. We are \( 90 \% \) confident that the population mean is between 10 and \( 14.8 \) 7. If we were to take one hundred samples from a population and calculate the \( 90 \% \) confidence intervals based on each sample, \( 90 \% \) of the time the population mean will be between 10 and \( 14.8 \) The mean price per gallon of regular gasoline is \( \$ 2.53 \) with a population standard deviation of \( \$ .45 \). Assume that you take a random sample of 36 gas stations and calculate the average price they charge for regular gasoline. a. (7) What is the probability that a randomly chosen sample of 36 gas stations will have a mean price of regular gas between \( \$ 2.25 \) and \( \$ 2.48 \) ? Please show all your work - including the graph with all significant values/abels, scales, and shaded area of interest. b. (2) Assume that another sample of 36 gas stations is selected and the mean price of regular gasoline is found to be \( \$ 2.90 \). Calculate the sampling error. c. (5) Is this sampling error unusual? Please justify your answer graphically and numerically as well as in a full sentence. 9. The police chief of River City reports that out of a total of 600 citations, a sample of 32 citations showed the mean amount of the fine to be \( \$ 82 \) with a standard deviation of \( \$ 7.4 \). Please show the formula used, values plugged in and your final response. a. (5) Construct an \( 80 \% \) CI for the mean fine on the traffic citations issued b. (1) State your conclusion in a full sentence c. (2) What does it mean that you are \( 80 \% \) confident? A group of statistics students decided to conduct a survey at their university to find the average number of hours students sleep per night. Please show the formula used, values plugged in and your final response. a. (3) Assuming a standard deviation of 1 hour, what is the required sample size for estimating the mean number of hours students sleep per night with a margin of error of \( 1 / 4 \) hour and a \( 90 \% \) level of confidence? A random sample of size 80 is taken from a population of size 600 with a population proportion of \( 0.46 \). a. (3) Calculate the expected value and the standard error of the sampling distribution of sample proportions. b. (5) What is the probability that another sample of 80 observations would have a sample proportion lower than \( 0.4 \) ? Please show all your work - including the graph with all significant values/labels, scales, and shaded area of interest. (3) Draw a sampling distribution of sample means and label on the graph both with the appropriate notation (when available) and in words: the standard error, a randomly chosen sample mean, the sampling error, and the margin of error for a CI built around the random sample mean.