4. Replacement times for TV sets are normally distributed with a mean of 8.2 years and a standard deviation of 1.1 years. (Change the final answer to a % and keep 2 decimal places) a) Find the probability that a randomly selected TV set will have a replacement time between 9.5 and 10.5 years. (Include diagram) b) Find the probability that 35 randomly selected TV sets will have a mean replacement time less than 8.0 years. (Include diagram)

The mean yearly snowfall for Boston over the past century is 43.5 inches. The standard deviation for this data is 19.4. Assume that the snowfall amounts are approximately normally distributed. If a random sample of 15 winters are selected at random, what is the probability that they will have a sample mean snowfall amount between 42.5 and 52 inches? Use 4 non-zero decimal places in your calculations. Round z-values to 2 decimal places.

Ex. A group of research scientists collect 2000 water samples from drinking water in Central Arizona. They test those samples for a certain chemical. The test isn’t 100% accurate. If the sample contains the chemical, it will show a positive test result 93% of the time. If the sample does not contain the chemical, it gives a negative result 97% of the time. If 170 groundwater samples contain chemicals, what is the probability the sample contains a chemical if you have a positive test return for the sample

A survey of 85 people in Kansas City gave the following results:

33 were fans of the Kansas City Chiefs

43 were fans of the Kansas City Royals

46 were fans of the Sporting Kansas City

23 were fans of the Chiefs and Royals

12 were fans of the Chiefs and Sporting KC

24 were fans of the Royals and Sporting KC

12 were fans of none of these teams.

How many people were fans of the Chiefs or Royals but not Sporting KC?

Consider the two processes below with specifications 100 plus or minus 10:

• Process A: mean of 100, standard deviation of 3
• Process B: mean of 105, standard deviation of 1
• n=5 for both processes

a.) Calculate Cp, Cpk, and Cpm and interpret the results

b.) What is the fraction non-conforming for each?

10.4 Comparing two means: Paired samples

"We want to know if there is a difference between the size of the shoe between mother and daughter, for which a sample of 10 pairs of mother and daughter is taken and a hypothesis test is performed."

1. State the hypotheses
2. what is the average value of the paired differences (d-bar)

3. Calculate the stadistic. T_calc

4. Do we accept or reject the null hypothesis?

4. the researcher wishes to use numerical descriptive measures to summarize the data on each of the two variables: hours worked per week and income earned per year.

1. Prepare and display a numerical summary report for each of the two variables including summary measures such as mean, median, range, variance, standard deviation, smallest and largest values and the three quartiles.                              

Notes: Use QUARTILE.EXC command to generate the three quartiles.

Compute the correlation coefficient using the relevant Excel function to measure the direction and strength of the linear relationship between the two variables. Display and interpret the correlation value.    

Data of Hours worked and yearly income as below

The dean of the a school has observed for several years and found that the probability distribution of the salary of the alumni’s first job after graduation is normal. The college collected information from 144 alumni and finds that the mean of their salary is $58k. Assuming a 95% confidence level, please do the following

1. Suppose the dean believes that the average salary of the population should be about $59k per year, with a standard deviation of $2k. We need to conclude that the mean salary is less than what the dean has believed to be:

(a) What are the null and alternate hypotheses ?

(b) What is the level of significance ?

(c) What is the standard error?

(d) Decide on the test statistic and calculate the value of the test statistic (hint: write the equation and calculate the statistic?

(e) What’s your decision regarding the hypothesis and interpret the result using test-score rejection region rule or p value rule.

1) A student wishes to determine the association between Hours Studied and Test Score. She also wants to determine the association between Hours of Sleep before the Test and the Test Score.

1. Which factor (hours studied or hours slept) is a stronger association with test score? (Show data to support this).

Hours Slept is a stronger association

2. What is the r-squared for hours studies vs test score and interpret it.

3. What is the regression equation for each association above and use the Linreg T test to determine whether the equations are valid predictors of Test Score.

1. Study vs Test score:
2. Sleep vs Test score

4. If someone studied 3.5 hours, what would be the predicted test score?                                         Is using your equation a valid way to predict this?

(a) Show that the sample variance s 2 = [Pn i=1(xi − x¯) 2 ]/(n − 1) can also be expressed as s 2 = [Pn i=1 x 2 i − ( Pn i=1 xi) 2 n ]/(n − 1). At a medical center, a sample of 36 days showed the following number of cardiograms done each day.

25 31 20 32 20 24 43 22 57 23 35 22 43 26 56 21 19 29 36 32 33 32 44 32 52 44 51 45 47 20 31 27 37 30 18 28

(b) (1 point) Find the sample mean ¯x and the sample variance s 2 x .

(c) (2 points) Construct a stem and leaf plot for the data and find the sample median.

(d) (3 points) Construct a 86% confidence interval for the population µ.

(e) (4 points) A researcher wishes to test the claim that the average number of cardiograms done each day is equal to or greater than 33. Is there evidence to support the claim at α = 0.05? Find the p-value.

(f) (1 point) Let x1, x2, · · ·, x36 be the data of 36 days of cardiograms above; let a and b be any nonzero constants. If y1 = a x1 +b, y2 = a x2 +b, · · ·, y36 = a x36 +b, and let ¯y and s 2 y be the sample mean and the sample variance of the yi ’s, respectively. What is the relationship between ¯x and ¯y? What is the relationship between s 2 x and s 2 y ?

(g) (5 points) Show that s 2 x is an unbiased estimator of the population mean σ 2 .

1. The Kenton Food company wished to test four different package designs for a new breakfast cereal. Twenty stores, with approximately equal sales volumes, were selected as the experimental units. each store was randomly assigned one of the package designs, with each package assigned to five stores. A fire occurred in one store during the study period, so this store had to be dropped from the study. Hence, one of the designs was tested in only 4 stores. The stores were chosen to be comparable in location and sale volume. Other relevant conditions that could affect sales, such as price, amount and location of shelf space, and special promotional efforts, were kept the same for all stores in the experiment.

Sales, in number of cases, were observed for the study period, and the results are recorded as follows.

Package Design #1: 11, 17, 16, 14, 15.

Package Design #2: 12, 10, 15, 19, 11.

Package Design #3: 23, 20, 18, 17, (fire)

Package Design #4: 27, 33, 22, 26, 28

Analyze the data and solve the hypothesis testing problem about equivalence of the package designs. Use α = .05.

It is thought that 12% of all students taking a particular course received a grade of A. In a sample of 155 students, it is found that 21 made an A. can we conclude that the ratio of students with grade of A is higher than 12%? To do so

a) State the null and alternative hypotheses.

b) Compute the test statistic-value.

c) Find the critical-value.

d) Identify the decision rule and express your decision.

The population mean and standard deviation are given below. Find the required probability and determine whether the given sample mean would be considered unusual.

For a sample of

n=70

find the probability of a sample mean being greater than

214

if

μ =213

and

σ =3.5

A license plate in a certain state consists of 4 digits, not necessarily distinct, and 3 letters, also not necessarily distinct.

(a) How many distinct license plates are possible if no restriction?

(b) How many distinct license plates are possible if it must begin and terminate by a digit?

(c) How many distinct license plates are possible if it must begin and terminate by a letter?

(d) How many distinct license plates are possible if the three letters must appear next to each other?

(e) How many distinct palindrome license plates are possible?
(A palindrome license plate is a license plate that reads the same from left to right as right to left)