# Statistical Analysis (estimation, distribution, etc.)

**Question description**

Question 4

Let x₁, x₂, …xₓ be a random sample of n i.i.d. observations following the Poisson distribution with parameter (mean) ϴ such that E(Xₐ) = ϴ. Consider three estimators of ϴ given as:

ϴ₁ = X, ϴ₂ = X₁, ϴ₃ = (x₁ + 2x₂)/3

i.) Show that all the three estimators are unbiased

ii.) Calculate which of three has the least Variance

Question 5

A psychology researcher wants to find out if exercising before taking a quiz affects a student’s performance. To test this he randomly assigns students to either exercise for 10 minutes before taking a short quiz, or to take the quiz without exercising first. 37 students exercise first and averaged 84% on the quiz with a standard deviation of 7% while 32 students skip exercising and score 81% with a standard deviation of 6%. Neither sample was substantially skewed.

a.) Write down the null and the alternative hypothesis in the test.

b.) Draw a picture of the samlpling distribution assuming that the null hypothesis is true, shading in the regions(s) that would result in a type 1 error (falsely rejecting the null hypothesis assuming that it’s true of 0.05

c.) Test the hypothesis at 95% confidence. NOTE: Instead of assuming that the variances are equal, first test for the equality of variances. Once you conclude regarding the equality of variances, test for the equality of mean performance.

d.) What is the P-value of the test?

Question 6

a.) For the IND ENG 716 group project, members of group 5 needed to determine if the percentage titanium content std. dev. In an aerospace casting is within the acceptable specifications of SD = .30. To do this, they tested the percentage content of titanium in 51 casting specimen and the sample std. dev. was 0.37. Construct the 95% confidence interval of the standard deviation and comment on whether the company’s castings are within the allowable specifications

b.) Next the group needed to study if the two ally impurity level detection tests that the company uses perform the same. To do this, they tested eight specimens on both apparatus and recorded the following impurity levels:

Is there evidence at 90% confidence that the tests differ in mean impurity levels? Hint: be careful to decide if these are homogenous samples or not, hence use the appropriate test. What is the P-value of your test?

Specimen |
Test 1 |
Test 2 |

1 |
1.2 |
2.0 |

2 |
1.3 |
1.7 |

3 |
1.5 |
1.5 |

4 |
1.4 |
1.3 |

5 |
1.7 |
2.0 |

6 |
1.8 |
2.1 |

7 |
1.4 |
1.7 |

8 |
1.0 |
1.6 |

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