MIS Exam 2

Central Limit Theorem

Bigger the samples, more normal and narrower distribution

Law of Large Numbers

As number of trials increase, actual value moves toward expected value

As sample size increases, observed stat will move toward actual (true) parameter

Confidence Intervals

A range of values, derived from sample data, that likely contains the true, unknown population parameter

Less about knowing and more about sensing

Confidence interval formula

Sample mean + Z score (s/quare root of n)

Degree of freedom formula

n-1

When would we use T Distribution

When n<30 and we don’t know sigma

When should we use z distribution

n is 30 or greater and know sigma

If DF > CL then

there is more uncertainty with smaller samples

Why do we need to make compromises when dealing with samples

• As we increase confidence, the confidence widens

• As we increase or decrease n, confidence interval lengthens

  • There’s natural tension between formulas

6 steps of Hypothesis Testing

1 - determine which of 4 facets of null correlate with problem

2 - Find alpha (usually given)

3- Calculate Test Stat

4 - Determine Current Value

5 - Make your test decision

If |TS| > |CV|

Reject the Null

If |TS|<|CV|

FTR null

(T/F) Failing to reject null does not mean its true

True

Type I Error

Reject true null (a)

Type II Error

FTR false null b

4 Facets of Null

1) status quo (normal state of events)

2) everything is unrelated

3) no difference between groups

4) everything arises from chance

Why can’t we subtract 1 - a to find b (beta)

Beta is part of a theoretical dataset that we don’t know - it’s actually a curve

Z test assumptions for sample 1 test

1) null is true

2) random sampling

3) central limit theorem is satisfied

4) interval ratio data

TS Formula

(Observed value - expected value)/ chance aka standard error of mean

What sign is used for Left tailed test

<

What sign is used for R tail test

>

What sign is used for 2 tailed test

not equal sign

SEM formula

s/ sqr root n

Why should we not put an = sign for all nulls?

Violates rule for being collectively exhausted (but since we’re aware we’ll put an = sign to make it easier)

Why is a (aka CV or “line of bs”) on such a low part of graph?

truly shows the reason for rejecting null

T Test sample instruction

1) null = true

2) random sampling

3) central limit theorem is satisfied

4) interval ratio data

What does the left side of the 2×2 matrix

No new action will be taken

What does the right side of the 2×2 matrix

New action will be taken

What are the rules of SED (Standard Error of Difference)

1) assume there are 2 population variances are not equal —> Non Pooled equation

2) If you know population is equal —> equal

3) n1 is close to n2 and s1=s2—> pooled

Degrees of Freedom formula

n1 + n2 - 2

When might there be dependent samples

1) 2 samples are paired or matched on related criteria (same shopper different store)

2) 1 sample is repeated measurement (before and after weightloss challenge)

P Value

Probability of attaining a value equal to or greater than the test stat. Given the test assumptions are true and a sound test structure

In a hypothesis test, if p-value < alpha we should....

Reject the null

What is\are the primary difference(s) between a one-tailed and two-tailed test?

1. one-tailed tests are used when there is reason to believe the results will fall either above or below the mean, while a two-tailed test are used when there is reason to believe the results could fall either above or below the mean.

2. one-tailed tests place all of alpha in either tail, while a two-tailed test tests evenly splits alpha into both tails

1 and 2

What is the key characteristic of a paired-difference test?

the 2 datasets are somehow related

or

the 2 datasets do NOT need to be of the same size

the 2 datasets are somehow related

If we create a 95% confidence interval…

there is no guarantee that the true population mean falls within our CI

we are 95% confident the true population mean falls within our CI

there is no guarantee that the true population mean falls within our CI

Which situation requires a one-tailed hypothesis test?

 

Group of answer choices

a company tests if a machine fills bottles with exactly 500 ml of shampoo

a researcher wants to know if a new process increases productivity

a  manager tests whether two departments have different satisfaction scores

a researcher wants to know if a new process increases productivity

The CLT requires the population to be normally distributed.

Group of answer choices

false

true

False

The paired-difference test can be used for --

A\ 2 samples that have been matched together based on some criteria
B| 1 sample with repeated measures.

C) Both

Which is true?

Both

Which of the following is true regarding the right tail of the distribution:

A\ As the Test Statistic increases, the p-value decreases.

B\  As the Test Statistic increases, the p-value increases

In a right-tailed test:

• The p-value is the area to the right of the test statistic.

• As the test statistic moves further right (increases), there is less area remaining to the right.

👉 So:

• Larger test statistic → smaller p-value

The -5.904 Test Stat indicates.

quiz scores decreased

quiz scores increased

✅ quiz scores increased

---

Why:

• The mean before = 5.3

• The mean after = 7.7 → clearly higher

The t-stat = -5.904 is negative because of how the difference was calculated (likely Before − After):

• Since After > Before → the difference is negative → t-stat is negative

👉 Negative t-stat here actually reflects an increase in scores.

It’s more important for females

Which model has a wider confidence interval

Model B

A\ The (shape of the) t-distributions accounts for the greater uncertainty we have when working with small populations.

 

B\ The (shape of the ) t-distribution accounts for the greater uncertainty we have when working with small samples.

Which of the above is true?

 

Group of answer choices

A

B

B

For the above test, the Observation values are female = 36 and the males = 31. 

thus, any test decision should not be used because of the unequal sample sizes

nonetheless, we can still use the test results, as the unequal sample size does not violate a test assumption

nonetheless, we can still use the test results, as the unequal sample size does not violate a test assumption

Which of the following is always an assumption of a hypothesis test?

Group of answer choices

the null is true

everything is related

none of the other choices

Null is true

In a hypothesis test, what is a type II error?

failing to reject a null that is really false

failing to reject a null that is really true

failing to reject a null that is really false

In step 3 of a 1 sample t-test -- the larger the Test Statistics the ____________ the difference between the sample mean and the expected value (population mean)

Group of answer choices

larger

smaller

• A larger t value means the sample mean is farther away from the expected value.

(larger)

For the above 2 tail hypothesis test, if the Test Statistics is represented by "X" we should fail to reject the null when...

A\ -1.96 < X <+1.96
B\ -1.64 < X < +1.64

A

In a one-sample Z-test for the means, the alternative hypothesis can be setup to test...

both tails

either the left or right tail

both of the other choices

both of the other choices

For the above 1 sample test of the means, the expected value of 14.50 is associated with the...

population

sample

population

In a one-sample Z-test for means, we are testing to see if the sample mean is...

equal to the population mean

not equal to the population mean

equal to the population mean

The above test compares upper-division, SDSU Fowler of Business College students to the population of California State university students. We can conclude the average number of units taken by the SDSU sample is similar to the population average of California State university students

 

Group of answer choices

true

false

true

Why:

• Test statistic: z = 0.38

• Critical values (two-tailed, α = 0.05): ±1.96

👉 Since 0.38 is between -1.96 and 1.96, we fail to reject the null hypothesis

If n > 30, the CLT states that the sampling distribution of the means will be...

Group of answer choices

normally distributed

skewed left

skewed right

normally distributed

For Model A, as n = 34, we should use which confidence interval:

A\ CI {14.185, 15.462}
B\ CI {14.162, 15,485}

A\ CI {14.185, 15.462}

The above test compares upper-division, SDSU Fowler of Business College students to the population of California State university students. We can conclude the average number of units taken by the SDSU sample ______the population average of California State university students

significantly exceeds

is significantly less than

is similar

significantly exceeds

Why:

• Sample mean = 15.89

• Population mean = 14.50
  👉 Sample is higher

---

Check significance:

• Test statistic: t = 3.57

• Critical value (two-tailed, α = 0.05, df = 8): ±2.31

👉 Since 3.57 > 2.31, we reject the null

In Step 3 of a 2-independent sample t-test, the T-statistic is the result of what measurement?

Group of answer choices

a comparison of the 2 sample means

a comparison of the 2 population means

a comparison of the 2 sample means

Which hypothesis is used when there is enough evidence to overthrow the status quo?

the null hypothesis

the alternative hypothesis

the alternative hypothesis

• The null hypothesis (H₀) represents the status quo

• When there is enough evidence to reject H₀, we support the alternative hypothesis (H₁)

In a hypothesis test our critical value is based upon --

1\ alpha, and
2\ the type of test we are doing (ex: Z or t), and
3\ whether we are doing a 1 or 2 tail test.

The above statement is ....

Group of answer choices

true

false

true

In a hypothesis test the p-value is a probability associated with obtaining the..

the Test Statistic

alpha

the Test Statistic

For the above test, the null and alternative was setup to run a 1-tail, left-tail test. What is the test decision?

Group of answer choices

reject the null

fail to reject the null

reject the null

As the sample size increases, the sample mean approaches the ...

Group of answer choices

population mean

the sample size

population mean