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Flashcards in Stat - Exam #2 Deck (90)
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31

What are the Estimators for population parameters?

Location = Mean, Median, Mode

Spread = Standard Deviation, Range, IQR

32

What are the properties of a good estimator?

1. Unbiased = expected value of estimator equals value of parameter;

2. Consistent = larger sample makes estimator more accurate;

3. Efficient = estimator has the smallest standard deviation

33

What is a Confidence Interval Estimate?

Range of values if an interval on the real number line;
-Expresses the natural uncertainty in the estimate

34

What is a Confidence Level?

The proportion of confidence intervals calculated from a large number of random samples that contain the value of the population parameter

Denoted = CL;
Common Values = 90%, 95%, 99%;
Decided by the researcher

35

What is the Significance Level?

The area outside of the region of confidence;

Denoted = sigma
Calculated = {1 - (CL/100)}

36

What is a Critical Value?

The pair of values that bound the region of confidence

Denoted = (+/- z_alpha/2’), (+/- t_alpha/2, n-1)

37

What is the Confidence Interval Estimate of a Population Parameter of a Population when Standard Deviation is KNOWN?

-An estimate of the value of a population parameter consisting of
1. an interval of number, and
2. a level of confidence that the interval contains the value of the population parameter

Denoted = CI% = (LCL,UCL)
LCL — Lower Confidence Limit; UCL — Upper Confidence Limit

38

What determines the WIDTH of the confidence interval?

Comes from the confidence level CHOSEN and the spread of the sample average

39

What is given by the confidence level?

Gives the area in the RIGHT tail (alpha/2), which gives the critical values bounding the region of confidence (+/-z_alpha/2)

40

Where does the CENTER of the confidence interval come from?

The CENTER of the confidence interval comes from the value of the SAMPLE AVERAGE (x-bar)

41

What are the steps of a Confidence Interval?

1. First find the critical values — defines the width of the interval (which is centered around the population mean, which is unknown)
2. Need to slide the interval over until it is centered on the sample average, which is known;
3. Convert the critical values into x-values;
4. Two x-values in the proper from make the confidence interval

42

What is the Margin of Error?

The RIGHT term in a confidence interval estimate;
-Determined the width of the confidence interval;
-Anything that changes the margin of error changes the width of the confidence interval

Margin = {z_(alpha/2)} x {sigma/(sq. rt of n)}

43

What happens with a REDUCED margin of error?

NARROWS confidence interval

44

What happens with an INCREASED margin of error?

WIDENED confidence interval

45

What are the 3 ways to change the Margin of Error?

1. Sample size: Increase = Narrow; Decrease = Widen;
2. Confidence level: Increase = Widen; Decrease = Narrow;
3. Standar deviation of the population: IMPOSSIBLE to change

46

What is the method for the Confidence Interval Estimate (z) of the Population Mean?

(Pop. SD is KNOWN)

1. Statistics = Sample Average (x-bar) & Population Standard Deviation (sigma);

2. Critical Value = Confidence Level (CL) & Critical Z-score (z_sigma/2);

3. Compute = CI% = {x-bar +/- (z_alpha/2) (sigma/sqrt. n)}

4. State = CI% = (LCL, UCL)

47

How is calculating confidence intervals estimates different when the population standard deviation is NOT known?

-Use the sample standard deviation (which can always be calculated) to calculate confidence interval estimates of a population parameter;
-Major difference is is the the DEGREES of FREEDOM must be known and must use the t-table to get critical values

48

What is the method for the Confidence Interval Estimate (t) of the Population Mean?

(Pop. SD is NOT known)

1. Statistics = Sample Average (x-bar) & SAMPLE Standard Deviation (s);

2. Critical Value = Confidence Level (CL), Degrees of Freedom (n-1) & Critical t-value {t_(alpha/2), (n-1)};

3. Compute = CI% = {x-bar +/- {t_(alpha/2), (n-1)} x [s/sqrt. n)]}

4. State = CI% = (LCL, UCL)

49

What is a Hypothesis Test?

1. First write 2 statement about a population parameter;
— Status quo value of the population parameter is given in the first statement;
— Claim made by the researcher is given in the second statement;
2. Sample data are collected and analyzed;
3. Finally concluded which statement is closer to the truth

50

How is a hypothesis test different from a confident interval calculation?

With a hypothesis test, you first must make some claim about a population parameter;
-Then collected data and determine if the claim is reasonable or not

51

What are the 3 steps of a hypothesis test?

1. Hypothesize = a set of hypotheses are written giving the status quo and the researchers claim;

2. Analyze = sample data are collected and analyzed;

3. Conclude = a conclusion as to which statement is closer to the truth is made

52

What is a hypothesis?

A statement about a population parameter;
EX: u=50 (pop mean)
- Can be for one or more populations
- Must be about the value of a population parameter (never a sample stat)
- Made BEFORE data is collected — and sample must be appropriate to the the statement;
**Two Types = Null and Alternative Hypothesis

53

What is a Null Hypothesis?

A statement that the population parameter has the status quo value;
- Denoted = “H-naught” (H0);
- EX = H0 : u = 72;
- Assumed TRUE in the hypothesis test until the sample evidence proves OTHERWISE;
-ALWAYS contains an EQUAL SIGN

54

What is an Alternative Hypothesis?

-A statement that a population parameter does NOT have the status quo value;
**Gives the researchers CLAIM;

- Denoted = “H-one” (H1);
- EX = H1 : u /= 72, H1 : u 72;
- Assumed FALSE in the hypothesis test until the sample evidence proves otherwise;
-NEVER contains an EQUAL SIGN

55

What is the purpose of analysis of a hypothesis test?

-Decide which hypothesis is closer to the truth, the null hypothesis or the alternative hypothesis;
-3 scenarios for a hypothesis test =
1. z-Test of the Mean
2. t-Test of the Mean
3. z-Test of Proportion

56

What are the 3 methods in each scenario to conduct a hypothesis test?

1. Critical Value Method = Traditional
2. P-value Method = Modern
3. Confidence Interval Method = Two-Sided

57

What is the conclusion to a hypothesis?

-Must chose only one of the two conclusions to end a hypothesis test;
1. REJECT the null hypothesis, or
2. NOT REJECT the null hypothesis
*Never enough information to prove a hypothesis is true, so a hypothesis can be shown to be false or not false — never shown true

58

What is Hypothesis Testing?

A producer that uses our knowledge of probability with evidence from a sample to test a claim about a characteristic of a population;
— Claim is about the value of a population parameter;
— Can be for one or more populations

59

What are the Assumptions of Hypothesis Testing?

1. Simple random sample;
2. Sample average is normally distributed

60

What is the logic behind hypothesis testing?

1. Assume status quo value (null) is TRUE (this is NOT confidence intervals);

2. Examine sample data;
— Sample evidence CLOSE to the status quo, support that value (null)
— Same evidence FAR from the status quo REFUTES that value and supports the alternative

3. Make one of two conclusions
— Status quo is reasonable given the sample
— Status quo is not reasonable given the sample