Slides 12 ⋆. 𐙚˚

Question 1

What is the Null Hypothesis?

Answer 1

A statement that there is no effect or no difference, serving as a default position in hypothesis testing

The Null Hypothesis is often denoted as H0.

Question 2

What is the Alternative Hypothesis?

Answer 2

A statement that contradicts the Null Hypothesis, suggesting that there is an effect or a difference

The Alternative Hypothesis is often denoted as H1 or Ha.

Question 3

What is the difference between a 1-Sided and 2-Sided test?

Answer 3

A 1-Sided test checks for an effect in one direction, while a 2-Sided test checks for an effect in both directions.

Question 4

What is a Significance Level (α)?

Answer 4

The probability of rejecting the Null Hypothesis when it is actually true, commonly set at 0.05.

Question 5

What does a p-value represent?

Answer 5

The probability of obtaining test results at least as extreme as the observed results, given that the Null Hypothesis is true.

Question 6

Is a good p-value sufficient for practical significance?

Answer 6

No, a ‘good’ p-value does not imply that the results are practically useful or interesting.

Question 7

How is the significance level (α) related to Type 1 and Type 2 errors?

Answer 7

A smaller α reduces the probability of a Type 1 error (false positive), while a larger α reduces the probability of a Type 2 error (false negative).

Question 8

When should you choose a smaller significance level?

Answer 8

When a false positive is especially dangerous or costly.

Question 9

What is the Central Limit Theorem (CLT) condition?

Answer 9

The condition that, with a large enough sample size, the sampling distribution of the sample mean will be approximately normally distributed.

Question 10

What is the t-distribution?

Answer 10

A probability distribution used when the sample size is small and population standard deviation is unknown, characterized by thicker tails than the normal distribution.

Question 11

What are the key characteristics of the t-distribution?

Answer 11

• Bell-shaped
• Centered at zero
• Has thicker tails than the normal distribution
• Defined by degrees of freedom (df = n - 1)

Question 12

When is the t-distribution nearly indistinguishable from the normal distribution?

Answer 12

When the degrees of freedom is about 30 or more.

Question 13

What is a t-test?

Answer 13

A statistical test used to compare the means of two groups when the sample size is small and the population variance is unknown.

Question 14

What is the significance level (α) for a typical hypothesis test?

Answer 14

Commonly set at 0.05.

Question 15

What is the average time for all runners who finished the Cherry Blossom Race in 2006?

Answer 15

93.29 minutes.

Question 16

What should you check for when determining normality in a sample?

Answer 16

Look for clear outliers and ensure adequate sample size.

Question 17

What is a key consideration when using the t-distribution with small samples?

Answer 17

The estimates become less precise, leading to potential issues when using the normal distribution for modeling.

Question 18

What is the formula for degrees of freedom in a t-distribution?

Answer 18

df = n - 1.

Question 19

What is the typical sample size needed to rely on the normal distribution?

Answer 19

Typically, a sample size greater than 30.

Question 20

What is the sample size for the Cherry Blossom Race study in 2017?

Answer 20

100 randomly sampled participants.

Question 21

What should you do if extreme outliers are present in your sample?

Answer 21

Reassess the normality condition as it may not be satisfied.

Question 22

In hypothesis testing, what does a two-sided test account for?

Answer 22

The probability of obtaining a more extreme value on both sides of the distribution.

Question 23

Fill in the blank: The p-value is different than the __________.

Answer 23

[size of the effect].

Question 24

What is the significance level α used in the hypothesis test?

Answer 24

0.05

This significance level is commonly used in hypothesis testing.

Question 25

What is the final p-value we will compare to our significance level?

Answer 25

0.0197 ## Footnote This p-value indicates the strength of the evidence against the null hypothesis.

Question 26

What is the conclusion if the p-value is smaller than alpha?

Answer 26

Reject the null hypothesis ## Footnote This indicates that there is statistically significant evidence to suggest a difference.

Question 27

What evidence does the data provide regarding the average run time for the Cherry Blossom Run in 2017 compared to 2006?

Answer 27

The average run time is different ## Footnote The observed value indicates that runners in 2017 were slower on average.

Question 28

What were the two hypotheses (H0 and HA) regarding traffic flow on Friday the 13th?

Answer 28

* H0: Average traffic flow on Friday 6th and 13th are equal. * HA: Average traffic flow on Friday 6th and 13th are different. ## Footnote These hypotheses were tested to investigate the impact of superstition on driving behavior.

Question 29

What is the sample size (n) of the traffic flow data collected?

Answer 29

10 ## Footnote A small sample size can affect the reliability of the statistical tests.

Question 30

What statistical distribution is used to model the sample mean when the sample size is small?

Answer 30

t-distribution ## Footnote The t-distribution is used instead of the normal distribution for small sample sizes.

Question 31

What does the Central Limit Theorem (CLT) state about sampling distributions?

Answer 31

They will be nearly normal if the population distribution is nearly normal. ## Footnote This is true regardless of sample size, but small samples are harder to verify for normality.

Question 32

What is the critical value used in constructing confidence intervals when using the t-distribution?

Answer 32

t* ## Footnote The critical value changes based on degrees of freedom and sample size.

Question 33

How is the margin of error (ME) calculated for confidence intervals?

Answer 33

ME = critical value * SE ## Footnote SE stands for standard error.

Question 34

What does a p-value of 0.0008 indicate in the context of the hypothesis test?

Answer 34

Strong evidence against the null hypothesis ## Footnote This low p-value suggests a significant difference in traffic flow.

Question 35

What is being investigated regarding people's behavior on Friday the 13th?

Answer 35

Whether people drive less compared to the previous week ## Footnote This is linked to superstitious behavior.

Question 36

What is the function used in R to calculate the p-value for the t-test?

Answer 36

pt() ## Footnote This function is specifically for the t-distribution.

Question 37

In what scenario should researchers be cautious when verifying the normality condition?

Answer 37

When working with small sample sizes ## Footnote Small samples can lead to unreliable assessments of normality.

Question 38

What is a confidence interval (CI) used for in hypothesis testing?

Answer 38

To estimate the true population difference ## Footnote CIs provide a range of values that likely contain the population parameter.

Question 39

How is the critical t-value determined for constructing a 95% confidence interval?

Answer 39

Using the qt() function with 0.975 as input ## Footnote This splits the 5% uncertainty for a 95% CI.

Question 40

What is the input value used for qt() when calculating a 95% confidence interval?

Answer 40

0.975

Question 41

In constructing a confidence interval for a small sample mean, what does the 5% uncertainty imply?

Answer 41

Half into the upper tail and half into the lower tail

Question 42

What is the average adult weight of an African elephant?

Answer 42

5 tons

Question 43

What was the sample mean weight of the 120 forest elephants in the study?

Answer 43

3.5 tons

Question 44

What was the standard deviation of the sample weight of the forest elephants?

Answer 44

1 ton

Question 45

Is there strong evidence that forest elephants are a different size based on the sample data?

Answer 45

This question requires statistical analysis to answer

Question 46

If Curry attempted 200 shots from 2-3 feet, what would we analyze?

Answer 46

How confident we could be that his shooting is better than the average NBA player

Question 47

What is the formula for standard error when calculating pHat?

Answer 47

sqrt(p*(1-p)/n)

Question 48

What is the value of Zstar1 used in confidence interval calculations?

Answer 48

1.96

Question 49

What is the value of Zstar2 used in confidence interval calculations?

Answer 49

2.576

Question 50

What is the formula for calculating the standard error of a mean (SE) given standard deviation (s) and sample size (n)?

Answer 50

s/sqrt(n)

Question 51

Fill in the blank: The formula for Tdiff is (1836 - 0) / ______.

Answer 51

SEdiff

Question 52

What is the sample size (n) used in the analysis of the forest elephants?

Answer 52

120

Question 53

What is the mean (mu) value used in the analysis?

Answer 53

5

Question 54

What is the sample mean (Xbar) of the forest elephants?

Answer 54

3.4

Question 55

What does SE represent in statistical analysis?

Answer 55

Standard Error

Question 56

True or False: The sample data suggests that forest elephants are larger than the average adult African elephant.

Answer 56

This requires further statistical evidence

Question 57

What is the formula used to calculate z in hypothesis testing?

Answer 57

(pHat - p) / SE

Question 58

What is the standard error (SE) for a sample proportion with p = 0.15 and n = 25?

Answer 58

sqrt(0.15*0.85/25)

Question 59

Which statistical software is mentioned for demonstrating the calculation of t*?

Answer 59

R