# Normal Probability, Confidence Interval & Hypothesis Testing

1- It has been reported that the mean income of parents of freshmen entering a particular university is $91,600. The president of a neighboring university feels that the mean income of the parents of his university's freshman class have a mean income greter than $91,600. The president selects 100 families randomly and finds the mean income to be $96,321 with a standard deviation of $9,555.

a) When analyzing this hypothesis, is this a two-tailed, right-tailed, or left-tailed test?

b) Using the results of this sample and a level of significance of .05, can we conclude that the he president of this neighboring university is correct in his feelings?

2- The length of human pregnancies is approximately normally distributed with a population mean of 266 days and a population standard deviation of 16 days.

a) What is the probability that a randomly selected pregnancy last less than 260 days?

b) What is the probability that a random sample of 20 pregnancies has a mean gestation period of less than 260 days?

3- The time for a Greased Lightnig oil change on a car is approximately normally distributed with an average of 17 minutes and a standard deviation or 2.5 minutes.

a) The "greased working in the pits guarantee that the oil change will take no longer than 20 minutes. If it takes longer than that, a customer will only have to pay half-fare. What is the approximate percentage of customers that can expect to pay half price?

b) If the Greased Lightning owner, Michael Blount (Chief Greaser), does not want to give this half fare price to more than 3% of its customers, how long should the company make the guarantee time limit?

4- A simple random sample of size "n" is drawn from a population that is normally distributed with a population standard deviation that is known to be 13. The sample mean, X, is found to be 108.

a) Construct a 95% CI using this data, giving the range in which the population mean is likely to be found if a sample of 25 elements is taken. (Again, show the data and the range, but you do not have to effect the full calculations.)

b) If the population standard deviation were not known, what can be used in its place (assuming we know that the population is normally distributed)?

c) If the population standard deviation is not known, what is the minimum size of the sample needed in order to use the normal distribution principles?

5 An arborist is interested in determining the mean diameter of mature white oak trees. He only has access to 7 trees. Using this sample, he determines that the sample mean is 49.09 cm and the sample standard deviation ("s") is 13.80.

Construct a 95% CI for the mean diameter of mature white oak trees using the above data.

6- For normal distributions, it is customary to contruct CI's at the 90%, 95%, and 99% levels. If we wanted to construct a CI at the 80% level, what would the z n multiplier be?

7- A two-lane highway with a posted speed limit of 45 miles per hour is located just outside a small hosing development. The residents are worried about the speed of the cards on the highway and want an estimate of the population mean speed of the cars on the highway. The transportation department of the township estimates that approximately 2500 cars traverse the highway daily during non-rush hours(when the speeds can be highter due to a lack of congestion).

This population is assumed to be normally distributed, with a population standard deviation ofo highway speed to be 8 miles per hour.

a) The township has the local high school take a sample of 12 cars during non-rush hour, selecting every 100th car until 12 cars had been measured.. The speeds measured were:

57.4 56.1 70.3 65.6

44.2 69.6 66.1 57.3

62.2 60.4 64.5 52.7

What is the point estimate of the population mean speed, using this sample data (a point estimate is the average of the numbers)?

b) Since we know that this point estimate is based on only one sample of 12 cars, we would like to find a range of speeds within which the true population mean is likely to fall. Construct a 90% confidence interval (CI) for this population mean. (do the calculation and the construction of the range of values.)

c) What is the maximum error of the mean for this CI (Calculate the number).

d) If the township were uncomfortable with this range of values, what would the multiplier be to increase the level of confidence to 99%.

e) For the 90% CI (part "b" above), what should the sample size be to limit the maximum error to within 2 miles per hour?

#### Solution Summary

The solution provides step by step method for the calculation of normal probabilities, confidence intervals and testing of hypothesis. Formula for the calculation and Interpretations of the results are also included. Interactive excel sheet is included. The user can edit the inputs and obtain the complete results for a new set of data.