PSM1_E2020SA 2020 Spring Moed A

Part I

Question 1

Analysis of each option:
:
Since and are independent, by the additive property of binomial distributions:

This is CORRECT.

:
where

• can only take even values:
• But can take all integer values:
• Therefore Y does NOT follow a binomial distribution

This is INCORRECT.

:

Therefore .
This is CORRECT.

:

This is CORRECT.

Question 2

We need to find where and are independent.

For the maximum to equal , at least one of or must equal .

Using the inclusion-exclusion principle:

Since and are independent:

For a binomial distribution :

Therefore:

Substituting back:

Question 3

We are given given lifetime with hours.
Since , we have:

For exponential distribution, the survival function is:

Therefore:

Question 4

We want to find for a sample of parts.

For exponential distribution with :

For the sample mean of independent observations:

By the central limit theorem, approximately. Therefore:

Part II

Question 5

According to confidence intervals:

Therefore the confidence interval is:

Question 6

For the confidence interval in Question 5, we used the large-sample formula:

The Central Limit Theorem allows us to use the normal approximation for the sample mean even when individual observations are not normally distributed, provided the sample size is sufficiently large.

Part III

Question 7

Error in original approach:
The calculation wrote but then calculated , which includes in the subtraction. This gives instead of .

Correct solution:

We want where .

Question 8

According to geometric distribution, if the probability no defects of either type is , then denoting as the number of tested items:

Question 9

If and were independent, then:

We know . Given that :

This means .

From the constraint :

Since , the maximum possible value of occurs when all probability is at :

This contradicts . Therefore, and are not independent.

Part IV

Question 10

We cannot start production based on this data alone, since the confidence interval for the difference in means includes values above five units. It is possible that the difference between the two means is less than five.

Question 11

According to the central limit theorem, the standard error difference.

Question 12

There is a fundamental relationship between hypothesis tests and confidence intervals:

For the two-sided test vs :

• If we reject at , then the confidence interval for will NOT contain
• If we fail to reject at , then the confidence interval for WILL contain

Since , we reject at the significance level.

Therefore, the confidence interval for does NOT contain zero.

Since the confidence interval doesn’t contain zero, it must be entirely on one side of zero - either entirely above zero or entirely below zero.

Part V

Question 13

The significance level is the probability of Type I error - rejecting when is true.

Since we reject when :

Therefore: for

Question 14

Power is the probability of correctly rejecting when the alternative hypothesis is true.

For the alternative :

Therefore: for

Question 15

The p-value is the probability of observing a result as extreme or more extreme than what was actually observed, assuming is true.

We observed defective parts. Since this is a one-sided test ():

Therefore: p-value for

Question 16

With imperfect detection (only 90% of defective parts are identified), the observed number of defective parts will be systematically lower than the true number.

Effect on Type I error:
Under (true ), we expect fewer observed defective parts due to the detection rate. This makes us more likely to observe , increasing the probability of incorrectly rejecting .

Effect on Type II error:
Under (true ), we also observe fewer defective parts, but this makes us more likely to correctly reject , thus decreasing the Type II error probability.

Part VI

Question 17

From the graph we can see the largest positive residual occurs at and . According to the linear regression, at this temp:

Therefore the residual is:

Question 18

If a different sample of observations was used to fit the model, we would not be surprised to get an estimate of the slope of the model of instead of .