A researcher wants to test if a six-sided die is balanced using a chi-squared goodness-of-fit test. If the die is rolled 60 times, what is the expected frequency for each face under the null hypothesis, and how many degrees of freedom (v) will the test statistic have?

Suppose that the cdf (cumulative distribution function) of 𝑋 is given by:  0 for 𝑎 < 0 ,  1 /5 for 0 ≤ 𝑎 < 2 𝐹(𝑎) = 2/5 for 2 ≤ 𝑎 < 4  and 1 for 𝑎 ≥ 4.

Determine Pr[a=4]

A student takes a multiple-choice exam. Suppose for each question they either know the

answer or gamble and choose an option at random. Further suppose that if they knows the

answer, the probability of a correct answer is 1, and if they gamble, this probability is 1/4.

To pass, students need to answer at least 60% of the questions correctly. The student has

“studied for a minimal pass,” i.e., with probability 0.6 they know the answer to a question.

For a single question, given that they answers it correctly, what is the probability that they

actually knew the answer (Pick the closest answer)?

Suppose you roll a fair 6-sided die 100 times (independently), and you get

$3 every time you roll a 6.

Let 𝑋1 be the number of dollars you win on rolls 1 through 25.

Let 𝑋2 be the number of dollars you win on rolls 26 through 50.

Let 𝑋3 be the number of dollars you win on rolls 51 through 75.

Let 𝑋4 be the number of dollars you win on rolls 76 throught 100.

Let 𝑋 = 𝑋1 + 𝑋2 + 𝑋3 + 𝑋4 be the total number of dollars you win over all 100 rolls.

Find the expectation of X

An engineer calculates the Pearson product-moment correlation coefficient between two variables, X and Y, and finds it to be r=−0.6. Based on this finding, what proportion of the total variation in the values of Y can be accounted for by a linear relationship with the values of X?

For

text

In

the given graph, how many intermediate vertices are required to travel from

node a to node e at a minimum cost?

The maximum value of 𝑥 such that the edge between the nodes B and C is included in every minimum spanning tree of the given graph is _________