# Normal Probability and Confidence Interval

1-The average time it takes to complete a test is first-year statistics is 46.2 minutes. the standard deviation is 8 minutes. Assume the times for taking the test are normally distributed. a) What is the probability that a randomly selected individual completes the test is less than 43 minutes. b) For a randomly selected group of 50 individuals, what is the probability that the mean test time (the mean of the sample) will be greater than 43 minutes? c) What is the probability that, if we select one individual randomly, they will complete the test in exactly 43 minutes?

2-The average age of a vehicle registered in the U.S. is 8 years (96 months). Assume the ages are normally distributed with a standard deviation of 16 months. a) If a car is selected randomly, what is the probability that its age is between 91 and 102 months? If a random sample of 36 vehicles is taken, what is the probability that the mean of their ages is between 91 and 102 months?

3- In the above problems involving samples, we have used the normal distribution probability table to calculate probabilities involving the means of the samples. In order to use this table, one of two criteria about the sample is necessary. What are these criteria? a) b) c) The proper name for the symbol "sigma/square root of n" is .

4- One grave digger can dig an average of 106 graves/week, with a sigma of 16.1 graves. If a sample of 35 grave diggers is chosen, what is the probability that the mean of the sample of graves dug would be between 100 and 110 per week?

5- I want to estimate the average money spent on cigarettes each week. I take a sample of 50 smokers and find the mean spent is $19 with a standard deviation of $6.80. a) For a 95% CI, estimate the interval within which the true population would fall. b) If I reduced the interval size to just 90%, what would that interval be? c) For the 90% CI, what is the maximum error of the estimate/mean?

6- I survey 30 individuals and I find that the average age of their snow blowers is 5.6 years. The sigma of this population is 0.8 years. a) Find the 90% CI within which the population mean falls. b) What would the 99% CI be?

7- a) I want to find the true population mean cost of a large plain pizza with a 95% level of confidence. (the sigma of the population is 0.26). How large should the sample if I want the cost to be accurate within $0.15 (maximum error of $0.15)?

b) I want to research the average monthly salary of teachers and want to be 90% confident that my estimate is correct. The sigma for theses salaries is $1,100.00. If I want the answer to be accurate within $150, how large should my sample be?

8- A small sample problem (sample size less than 30- a "t-distribution.") 10 cars were selected at random and the depth of their oil was measured. The mean for these 10 cars was .32 inches with a sample standard deviation "s" of .08 inches. Find the 95% CI of the population mean depth. (Assume the variable- the depth- is approximately normally distributed). What would the multiplier be for a 99% CI?

#### Solution Summary

The solution provides step by step method for the calculation of confidence interval, sample size, margin of error and probability using the Z score. Formula for the calculation and Interpretations of the results are also included.