PSM1_E2020SB 2020 Spring Moed B

Part I

Question 1

According to a hypergeometric distribution:

Question 2

Assertion I - incorrect.
Assertion II - correct.

Question 3

The p-value is calculated based on the test statistic and the sampling distribution under the null hypothesis. It represents the probability of observing a test statistic as extreme or more extreme than what was actually observed, assuming is true.

The significance level is simply the threshold we choose for making a decision about rejecting . The p-value calculation does not depend on our choice of .

Since the same data produces the same test statistic, and the same test statistic produces the same p-value regardless of what decision criterion we use, changing from to will not change the p-value.

What changes is only our decision:

• With : p-value , so do not reject
• With : p-value , so reject

Question 4

According to poisson distribution:

Question 5

The probability of a product to be found with no scratches is:

Using a negative binomial:

Question 6

We are given , and repair times are for scratch, for scratch.

First, find probabilities for each repair scenario:

For each product, the expected repair time:

The variance:

For products:

Using normal approximation:

Therefore:

Part II

Question 7

Given: confidence interval , , .

For a confidence interval with unknown :

From the confidence interval:

The margin of error equals:

Substituting known values:

Solving for :

Question 8

There is a direct relationship between confidence intervals and hypothesis testing:

• A confidence interval corresponds to a two-sided test with
• If the null hypothesis value is inside the confidence interval, then do not reject (-value )
• If the null hypothesis value is outside the confidence interval, then reject (-value )

Given:

• confidence interval:
• Null hypothesis:

Since lies within the interval , we would not reject .

Therefore, the -value .

Question 9

Assertion I is correct.
Assertion II is incorrect.

Question 10

The four small horizontal lines shown on the graph show average one sample standard deviation for each machine (with standard deviation estimated separately for each machine)

Question 11

We cannot start production, since the whole confidence interval for the difference in means is above one unit, indicating that the difference between the two means is greater than one unit.

Question 12

From the JMP output:

• Test statistic:
• Degrees of freedom:
• The one-sided p-value is directly provided as ""

Since the observed difference () is positive and we’re testing ₁₂₁, we need the right tail probability:

-value

Part III

Question 13

From the regression output:

For temperature = , the predicted weight is:

Therefore the residual for the observation with and is:

Question 14

The standard error is an estimate of the standard deviation of the estimate of the slope of the fitted line.

Question 15

A composite sample is contaminated if at least one of the individual samples is contaminated.

Using the complement:

Therefore:

Question 16

Cost scenarios per week:

• If composite not contaminated: shekels
• If composite contaminated: shekels

From the previous question:

Expected weekly cost:

The annual expected cost: .

Therefore the annual savings:

Question 17

For a binomial distribution, we need: fixed number of independent trials, each with two outcomes and constant probability of success.

Analyzing each option:

• Options a, b, e: The number of tests varies based on contamination results
• Option c: Always exactly tests (constant, not random)
• Option d: Exactly independent trials (weeks), each with probability of contamination

Therefore, option d. is the correct answer.

Question 18

Cost less than in the original version above.