Statistics and Probability Answers

Drinking water has become an important concern among people. The quality of drinking water

must be monitored as often as possible during the day for possible contamination. Another variable of

concern is the pH level, which measures the alkalinity or the acidity of the water. A pH below 7.0 is

acidic while a pH above 7.0 is alkaline. A pH of 7.0 is neutral. A water-treatment plant has a target pH

of 8.0. Based on 16 random water samples, the mean and the standard deviation were found to be: X¯= 7.6 s = 0.4

Does the sample mean provide enough evidence that it differs significantly from the target

mean? In other words, does the sample come from a population whose mean is the same as the

target pH of ? Use , two-tailed test.

1.    Assuming that the samples come from normal distributions, find the margin of error  given the following:

a. n = 10 and X = 28 with s = 4.0, 90% confidence

b. n = 16 and X = 50 with s = 4.2, 95% confidence

c. n = 20 and X = 68.2 with s = 2.5, 90% confidence

d. n = 23 and X = 80.6 with s = 3.2, 95% confidence

e. n = 25 and X = 92.8 with s = 2.6, 99% confidence

2.    Using the information in number 2, find the interval estimates of the population mean.

A researcher used a developed problem solving test to randomly select 50 Grade 6 pupils. In

this sample, and . The mean and the standard deviation of the population used in

the standardization of the test were 75 and 15, respectively. Use the 95% confidence level

A manufacturer of a flu vaccine is concerned about the quality of its flu serum. Batches of serum are processed by three different departments having rejection rates of 0.20, 0.08, and 0.21, respectively. The inspections by the three departments are sequential and independent.

(a) What is the probability that a batch of serum survives the first departmental inspection but is rejected by the second department?

(b) What is the probability that a batch of serum is rejected by the third department?

A manufacturer of a flu vaccine is concerned about the quality of its flu serum. Batches of serum are processed by three different departments having rejection rates of 0.20, 0.08, and 0.12, respectively. The inspections by the three departments are sequential and independent.

(a) What is the probability that a batch of serum survives the first two departmental inspection but is rejected by the third department?

(b) What is the probability that a batch of serum is rejected by the first department?

A manufacturing firm employs three analytical plans for the design and development of a particular product. For cost reasons, all three are used at varying times. In fact, plans 1, 2, and 3 are used for 40%,, 15%, and 45% of the products respectively. The "defect rate:" is different for the three procedures as follows:

P(D\P1)=0.01, P{D\P2) = 0.03. P(D|P3 ) = 0.02,

where P(D\Pi) is the probability of a, defective product, given plan i. If a random product was observed and found to be defective, which plan was most likely used and thus responsible?

Suppose a car rental firm wants to estimate the average number of kilometers traveled per day by each of its cars rented in a certain city. A random sample of 20 cars rented in that city reveals that the sample mean travel distance per day is 85.5 kilometers, with a population standard deviation of 19.3 kilometers. Compute a 99% confidence interval to estimate Q. (2 points) Interpret your answer. (1 point)​

If F(x) 1/39(3x-2)2;0<_x_<3

O: elsewhere

1. Verify that F(x) is a PDF

2. find E(x) and Var(x)