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61

How do you define “close” and “far” from the status quo?

-Using the properties of the normal curve;
-Determine the critical z-score (remember that z-score of a point is just how many standard deviations away from the mean)
— Any value CLOSER to the mean than the z-score is CLOSE;
— Any value further way from the mean than the z-score if FAR
*Z-scores are then converted to x-values that support or refute the null hypothesis

62

95% Level of Confidence

-Any value inside Z-scores of +/-1.96 is CLOSE to the mean and support the null;
-Any value outside Z-scores +/-1.96 is FAR from the mean and refutes the null hypothesis (reject the null)

63

What are the 3 situation scenarios in hypothesis testing?

1. Two-tail situation;
2. One-tail situation to the left, and
3. One-tail situation to the right

64

Two-Tail Situations

The null hypothesis can be rejected by sample evidence that is too big OR too small (rejection in both tail regions)
-Two- tail hypothesis: H0: u=72; H1 u /= 72

65

One-Tail Situations

The null hypothesis can be rejected by sample evidence that is ONLY too big or too small (rejection in ONLY ONE tail region)
-Right-Tail hypothesis: H0: u= 72; H1: 72< u
-Left-Tail hypothesis: H0: u = 72; H1 u <72

66

What is a Type 1 Error?

The null hypothesis is TRUE; but we REJECT the null hypothesis in the test;
-Denoted “alpha"

67

What is a Type II Error?

The null hypothesis is FALSE, but we do NOT REJECT the null in the hypothesis test;
- Denoted “beta"

68

Which type of Error is stats most concerned with?

-Type I;
-Choose the probability of making a Type 1 Error early in the hypothesis test;
-Usually let a Type II error float to whatever it becomes;
**Type 1 Error = Level of Significance

69

What is Level of Significance?

The probability of making a Type I Error;
-Denoted “alpha”
-Rejection region in the hypothesis testing

70

How do you choose the level of significant?

-If the consequences of making a Type I error are severe, choose the level of significance to be SMALL (alpha = 0.01);
-If the consequences are NOT severe, the level of significance should be larger (alpha = 0.05 or 0.10);
*Inverse relationship of Type I and Type II errors;
*Raise probability of Type I (raise alpha), reduces the probability of a Type II error

71

How do Significance Level and Confidence Level relate?

-Like two side of a coin;
-A 5% significance level means a 95% confidence level of giving the correct conclusion

72

What is Level of Confidence?

-The probability of NOT making a Type I Error;
- Calculation: 1-alpha

73

What are the 3 methods of conducting a hypothesis test about a KNOW population mean using z-scores?

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

74

What is a Critical Value?

*Critical Value Method (z) =
A z-score which is critical to separate the REJECTION REGION from the ACCEPTANCE REGION;

-Denoted: +/- z_alpha/2, -z_alpha, +z_alpha

75

What is the Rejection Region of the Critical Value?

The set of all z-scores that are FAR from the mean, such that a NULL hypothesis is REJECTED;

- Denoted: alpha;
-Sometimes called the Critical Region

76

What is a Test Statistic?

A z-score, calculated form sample data, which is used to test if the NULL hypothesis is closer to the truth;
-Denoted: z_0;

-Calculation: z_0 = {(sample mean - pop. mean)/(pop. SD/ sqrt of sample (n))}

*SAME calculation for Left, Right, and Two-Tail Critical Values

77

Left-Tail Critical Value Method

1. Hypothesis:
H0: u = u0;
H1: u < u0;

2. Critical Value = -z_alpha;

3. Calculation

4. Reject: z_0 < -z_alpha;

5. Conclusion: Do, or do not, reject null

78

Two-Tail Critical Value Method

1. Hypothesis:
H0: u = u0;
H1: u NOT equal u0;

2. Critical Value = +/-z_alpha/2;

3. Calculation

4. . Reject: z_0 < -z_alpha/2 OR z_alpha/2 < z_0;

5. . Conclusion: Do, or do not, reject null

79

Right Tail Critical Value

1. Hypothesis:
H0: u = u0;
H1: u > u0;

2. Critical Value = +z_alpha;

3. Calculation

4. . Reject: z_alpha < z_0;

5. Conclusion: Do, or do not, reject null

80

When can you used the Critical Value method?

-Method is ROBUST for small deviation from normality (use normal probability plot),

- But NOT robust for data with outliers = use boxplot

81

What is a P-Value?

The probability of repeating an experiment under the assumption that the null hypothesis is true and getting a test statistic as extreme, or more extreme, than the value observed;

-This is the area under the curve from the TEST STAT to INFINITY;
-In one tail if is is a one-tail situation and in both tails if it is a two-tail situation

82

What is the P-Value method?

1. Test Stat for Left-Tail, Two-Tail, and Right-Tail = z_0;

2. Extreme Values:
-Left Tail = z < z_0;
-Two Tail = (z < -z_0) + (+z_0 < z);
-Right Tail = z_0 < z

3. Calculate = (Area from Z-table) X (Number of Tails)

83

What are the advantages of using the P-Value OVER the Critical Value Method?

1. The decision made in the P-value method is the SAME way every time — no need to look up a different P-Value every time
2. P-value gives info about the STRENGTH of evidence; P-value close to the level of significance means that the evidence for making conclusion is WEAK; P-value fat from the level of significance means that the evidence for making a conclusion is STRONG

84

How do you determine the hypothesis using a P-Value?

P-Value GREATER than alpha = DO NOT Reject the Null;

P-Value LESS than alpha = REJECT the Null

85

When are Confidence Intervals used?

ONLY when you have a two-tail hypothesis test;
-Because rejection region in a confidence interval is ALWAYS in both tails

86

What must be used to test claims about a population mean when the population standard deviation is NOT known?

-Sample standard deviation is always known (or can be calculated) and use the *t-table* (NOT the z-table)

87

What is a Critical Value?

-a t-value which is critical to separated the rejection region from the acceptance region;

-Denoted: {(+/- t_alpha/2), (n-1, -t_alpha), n-1, (+t_alpha, n-1)}

88

What is a Test Statistic for hypothesis tests when the population standard deviation is NOT known (using t-test)?

-a t-value, calculated from SAMPLE data, which is used to test is the NULL hypothesis is closer to the truth;
-Denoted: t_0;

-Calculation: t_0 = (sample mean - pop. mean)/(s/ sqrt. n)

(s = sample standard deviation; n = sample size)

89

What is the J-Method to Find Area (t)?

*Method ONLY for the t-table;
1. Start in the left margin at the degrees-of-freedom;
2. Go across the row until you find the number closest to the value of the test stat;
3. Read area in one tail at the top of the column

90

How do you decide the null using the t-table once the P-value is determined?

-One the P-value is obtained, the decision is made the SAME way as when the population standard deviation is KNOWN