Statistics and Probability Answers

Suppose an engineer is 95% confident that the probability of rejecting a product is going to be 0.5 ± 0.2.

1. Use this information to construct a beta prior for θ.

Hint. Use normal approximation for the beta distribution. Note that if X ∼ Beta(α,β), then and.

1. Suppose that the engineer observed 4 rejections among 10 products. Find the posterior distribution for θ using the prior you proposed in (a).

Problem 2. Suppose the posterior distribution of θ is given by a mixture normal distribution: θ|data ∼ 0.5 · N(0,1) + 0.5 · N(10,1).

Find the

1. 95% HPD credible interval of θ.
2. 95% equal-tail credible interval of θ

1. The lifetimes of the light bulbs produced by Filip Company are normally distributed with a mean of 1,150 hours and a standard deviation of 175 hours.

• What is the probability that a randomly chosen light bulb will have a lifetime of more than 1,000 hours?
• What percentage of the light bulbs would be expected to last between 1,000 and 1,500 hours?

A normally distributed random variable X has a mean of 500 and a standard deviation of 40.

(iii) Suppose that Dominic serves 101 times. The sequence of serves is {Left, Right, Left, Right … Left, Right, Left, Right, Left}. Explain why Dominic’s strategy might not be consistent with equilibrium behavior. Describe a statistical test you might use. (iv) What is a randomized t test and why did some authors need to use such a test? 5. (i) Some authors think a hot hand results from positive serial correlation. Others think it results from non-stationarity. Explain briefly the difference between those two concepts. (ii) Explain why it may be easier to find a hot hand in baseball than in field goal shooting in basketball. (iii) Explain what the following table shows. To what does the bias adjustment refer?

(iv) Explain what columns (1) and (5) of the following table show. 

It is known that the weights of mangoes harvested in a farm are normally distributed with the mean of 220 grams and a standard deviation of 25 grams.

1. (i) Interpret the coefficients in the final two columns of the following table. Explain (briefly) why the results here suggested there was an inefficiency. (ii) A tennis player’s serve lands in with probability x. Conditional on the serve being in, the server wins the point with probability 0.9025 − �2 3 . Find the optimal value for the second serve, x2. Show that the optimal value for the first serve, x1, is smaller than x2. Explain why this occurs. (iii) Juan has a batting average of .350 with runners in scoring position (i.e., a teammate is on second and/or third base). He has a batting average of .300 when there are no runners in scoring position. Fernando has a batting average of .340 with runners in scoring position and .290 with no runners in scoring position. What can we conclude about Juan’s overall batting average relative to Fernando’s? (iv) Explain what Abramitzky et al. found concerning success rates for line call challenges. 

In a survey of insect life near a stream, a student collected data about the number of

different insect species (y) that were found at different distances (x) in meters from the

stream.

Distance (x) 2 5 8 11 14 17 22 33 39

Insect species (y) 26 25 19 19 14 9 5 3 2

a) Draw a scatter diagram to show the data and describe the correlation between the

number of different insect species and the distance from the stream.

b) Find the Spearman’s rank correlation coefficient and interpret.

Suppose (Ω, 𝒜, 𝑃) is a probability space and 𝐵 is an event with 𝑃(𝐵) > 0. Prove that (Ω, 𝒜, 𝑃(⋅ |𝐵)), where 𝑃(⋅ |𝐵) is the conditional probability given 𝐵, is a probability space.

Please provide step by step solutions and logical explanations :)

F(x,y)={k(1-x^2) find the value of k

Suppose an engineer is 95% confident that the probability of rejecting aproduct is going to be 0.5±0.2.

(a) Use this information to construct a beta prior forθ.Hint.Use normal approximation for the beta distribution.

Note that if X∼Beta(α,β),thenE[X] =α/(α+β) andV ar(X) =αβ/((α+β)^2(α+β+1)).

(b) Suppose that the engineer observed 4 rejections among 10 products. Find the poste-rior distribution for θ using the prior you proposed in (a)