Statistics and Probability Answers

lTo determine if a single die is balanced or fair, the die was rolled 600 times. The observed frequencies with which each of the six sides of the die turned up are recorded in the following table:
Face 2 3 4 5 6
observed frequency 114 92 84 101 107 102
Is there sufficient evidence to conclude, at the %5 level of significance, that the die is not fair?

Let Y,X have joint pdf
f xy( x,y)={ 4 ; 0 smaller than equal to x smaller than equal to 1 , 0 smaller than equal to y smaller than equal to 1
and 0 ; otherwise
Find fx ( x) , fy ( y ) ,fx/y ( x/y) ,f y/x ( y/x ) also find E ( X/ Y = y) and E (Y /X = x)

Let X1,X2,............Xn be a random sample from a distribution with density function
f(x, alpha)={ 1/ alpha ; 0 smaller than equal to x smaller than equal to alpha
and 0 ; otherwise
Obtain the maximum likelihood estimator of θ

Let X1,X2,............Xn be a random sample with E(Xi)=m and var( Xi)= sigma^2 for all I= 1,2.....n . show that S^2=1/n-1 sum ( Xi-mean) is an unbiased estimator of sigma^2.

Find the mean and standard deviation of the following frequency distribution. (5) Class Intervals Frequency 6.45 – 7.25 7.25 – 8.05 8.05 – 8.85 8.85 – 9.65 9.65 – 10.45 10.45 – 11.25 11.25 – 12.05 12.05 – 12.85 12.85 – 13.65 7 11 34 67 62 67 33 13 6 Total 300

A student obtained the following results for the normality of a Nitric acid solution: 0.424; 0.528; 0.621; 0.433; 0.537. Should 0.621as a result be rejected statistically?

In a study on the physical activity of young​ adults, pediatric researchers measured overall physical activity as the total number of registered movements​ (counts) over a period of time and then computed the number of counts per minute​ (cpm) for each subject. The study revealed that the overall physical activity of obese young adults has a mean of mu equals 325μ=325 cpm and a standard deviation of sigma equals 90σ=90 cpm. In a random sample of n equals 100n=100 obese young​ adults, consider the sample mean counts per​ minute, x overbarx.

1. mu bar-x =

The ages of the employees of a fruit-juice outlet are as follows: 20,18,21,20,65,19,19 i) Compute the mean, the median and the mode of the ages. ii) How would these three measures of central tendency be affected if the oldest employee retired? Also, find the coefficients of variation in this case.

2.45 Construct a cross-tabulation table using cola preference (Rola or Koka) as the row variable and Rola-Cola purchase history (Yes or No) as the column variable. Based on the table, answer the following.

a How many shoppers who preferred Rola-Cola in the blind taste test had previously purchased Rola-Cola? 19
b How many shoppers who preferred Koka-Cola in the blind taste test had not previously purchased Rola-Cola? 21
c What kind of relationship, if any, seems to exist between cola preference and Rola-Cola purchase history?

Why do we reject null hypothesis when the Chi square statistic is less than the tabulated value of chi-square at say 5% level of significance and accept when it is more?

What does it mean to have a Chi-square value more or less than the value assigned to a certain level of significance?