Part I

Question 1

We know that . Therefore, using a geometric distribution:

Question 2

First, we notice that:

Now we need to find . According to conditional probability:

Question 3

According to poisson distribution:

Question 4

First, we notice that:

Therefore, using the multiplication principle for independent events, the probability of this specific scenario is:

Question 5

Since any one of the total games could be the one with exactly ripped jerseys, we need to account for all possibilities using the multiplication principle:

Question 6

We are asked what is .
First, we notice that:

Therefore, assuming they are independent:

Question 7

The probabilities for each jersey being defective are different this time - we need to use hypergeometric distributions:

Therefore, assuming they are independent:

Part II

Question 8

We need to find . Because we know it has a normal distribution with and , we can use the standard normal curve:

Question 9

Using a binomial distribution and our previous result:

Question 10

We denote:

- no fine.
- small fine.
- large fine.

We already found that for a single project .
Furthermore:

Calculating the expected fine for each case and then summing them up yields:

The variance:

Which means:

Therefore, the expected fine for projects is:

And the variance:

Question 11

Using the previous answer and the standard normal curve:

Part III

Question 12

The confidence interval is:

Question 13

We know that doesn’t sit inside the confidence bounds. Therefore, we gather that is rejected, and .

Question 14

Assuming stays the same, the observation doesn’t affect the width of the sample, as it is determined by our choice of , the standard deviation , and the sample size .

Question 15

The new confidence interval will be wider, as can be seen from the confidence interval definition.

Part IV

Question 16

Simply reading the table gives:

Question 17

For a confidence interval for the slope, we use:

From the statistical output:

Slope estimate
Standard Error
Degrees of freedom
For CI:

Therefore, the confidence interval is:

Question 18

We would not expect the following to the variance of the points above and below the fitted line will be smaller

Part V

Question 19

According to error types, The null hypothesis will not be rejected and we might be making a Type II error.

Question 20

For this two-stage protocol, we need to find the overall Type I error rate.

Under (null hypothesis is true), we reject if:

We reject at Stage 1 (probability = ), OR
We fail to reject at Stage 1 AND reject at Stage 2

For Stage 2 to occur:

Must first NOT reject at Stage 1: probability =
Then reject at Stage 2: probability =
Combined probability =

Overall Type I Error:

פנגייה

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