Data Distribution Questions and Answers

ABC chemical plant supervisor is concerned about the high reaction times, y of a batch process. He wants to examine the root cause of the problem and select a sequence of 6 previous batches at different concentration of impurities, x. From these data the following are calculated.
${\displaystyle \sum x=200}$
${\displaystyle \sum y=436}$
${\displaystyle \sum xy=12515}$
${\displaystyle \sum x^2=7250}$
${\displaystyle \sum y^2=39234}$
${\displaystyle n=6}$
1) Compute the value of the correlation coefficient between the y and x for the given data.
2) Interpret the result obtained from part (i).
3) Comment on the suggestion that high reaction times are caused by concentration of impurities.

As an engineer working for a water bottling company, you collect the following data in order to test the perfomance of the bottling systems. Plot the data and calculate the mean and standard deviation.
$$\begin{array}{|cc|}\hline \text{Milliters of Water in the Bottle}& Frequency\\ 485& 13\\ 490& 17\\ 495& 25\\ 500& 40\\ 505& 23\\ 510& 18\\ 515& 15\\ \hline\end{array}$$

Use the following data set : 7 , 17, 30, 18, 24, 16, 3, 18,7 to answer questions
The midrange of the data
Select one:
a. 27
b. 16.5
c. 20

Use the following data set: 13, 8, 10, 7, 13, 9, 7, 11, 12, 10, 13 to answer questions
The range of the data
Select one:
a. 10
b. 6
c. 20

Cameron is birdwatching at the coast. He has seen 3 gulls and 6 other birds. Considering this data, how many of the next 18 birds Cameron sees would you expect to be gulls?

Use the Sample Variance Definition to find the variance and standard deviation of the data. Round the answers to one decimal place.
8, 7, 3, 2, 10, 4

Consider the data: 12, 9, 11, 15, 13, 15, 16 (Show your work.)
a) Determine the mean.
b) Determine the standard deviation.
c) Determine the median.

FIND OUT P VALUE FOR GIVEN DATA
TEST STATISTICS T SCORE -2.3698
SAMPLE SIZE ${\displaystyle =100}$
THE TEST IS TWO TAILED TEST

Define sample space of following experiments:
1. Number of bit errors in 6-bit data.
2. A random number generated between zero and one.

Find the mean, median, and mode of the following data:
(round to 1 decimal place as needed)
35,12,17,30,12,29,33,12,29,15
Mean=?
Median=?
Mode=?

Find the mean of the following data: 12,10,15,10,16,12,10,15,15,13

Construct all random samples consisting three observations from the given data. Arrange the observations in ascending order without replacement and repetition.
86 89 92 95 98

Construct your own table showing probability distribution using 8 data. Solve for mean, variance and standard deviation of the distribution.

Provide an appropriate response.
Find the equation of the regression line for the given data.
$$\begin{array}{|ccccccccccc|}\hline x& -5& -3& 4& 1& -1& -2& 0& 2& 3& -4\\ y& 11& 6& -6& -1& 3& 4& 1& -4& -5& 8\\ \hline\end{array}$$

In a dataset, X is the independent variable, and Y is the dependent variable. Which of the following statement is correct?
1) Scatter plot between X and Y helps to see the pattern of the data spread.
2) Scatter plot between X and Y helps to find the regression line.
3) None of these
4) Scatter plot between, X and Y helps to find the value of r.
In a dataset, X is the independent variable, and Y is dependent variable, and r is Karl Pearson's coefficient of correlation. If ${\displaystyle r=0.099}$ then
1) None of these
2) there is a weekly positive correlation between X and Y.
3) There is no correlation between X and Y.
4) There is a strongly positive correlation between X and Y.

Three cards are drawn without replacement from the 12 face cards (jacks, queens, and kings) of an ordinary deck of 52 playing cards. Let X be the number of kings selected and Y the number of jacks.
a) Find the joint probability distribution of X and Y;
b) Find the marginal probability distributions of X and Y.
c) ${\displaystyle P[(X,Y)\in A]}$, where A is the region given by ${\displaystyle \{(x,y)\mid x+y\ge 2\}}$.

Based on available data, the distribution function chosen to model particular data is
${\displaystyle f_x\left(x\right)=\lambda e^{-\lambda x}}$
Given these three data points: 3.6, 5.5, and 3.8, use the Maximum Likelihood Estimation method to determine the value of ${\displaystyle \lambda }$ to 3 decimal places.

A random sample of 25 of the 400 member of the Bigtime Theater Company is surveyed about how may plays each has acted in. Which statement below would be an appropriate inference about the data?
3,5,5,3,4,4,1,3,6,10,1,3,4,5,1,2,4,2,3,2,5,5,5,5,6
1) Most members have acted in 2 plays.
2) Most members have acted in more than 6 plays.
3) Most members have acted in between 1-4 plays.
4) The number of members who have acted in 10 plays is greater than the number of members who have acted in 1 play.

The average price for a gallon of gasoline in the United States is $3.73 and in Russia it is $3.40 (Bloomberg Businessweek, March 5 March 11, 2012). Assume these averages are the population means in the two countries and that the probability distributions are normally distributed with a standard deviation of $.25 in the United States and a standard deviation of $.20 in Russia.
What is the probability that a randomly selected gas station in the United States charges less than $3.50 per gallon?
What percentage of the gas stations in Russia charge less than $3.50 per gallon?
What is the probability that a randomly selected gas station in Russia charged more than the mean price in the United States?

Let X denote the data transfer time (ms) in a grid computing system (the time required for data transfer between a “worker” computer and a “master” computer.
Suppose that X has a gamma distribution with mean value 37.5 ms and standard deviation 21.6 (suggested by the article “Computation Time of Grid Computing with Data Transfer Times that Follow a Gamma Distribution,” Proceedings of the First International Conference on Semantics, Knowledge, and Grid, 2005).
a. What are the values of a and b?
b. What is the probability that data transfer time exceeds 50 ms?