Proving (or disproving) the Null Hypothesis

1. From the sample and hypothesis benchmark we calculate a test statistic

2. From the hypothesis and degree of confidence we calculate a critical region, bounded by critical values

3. If the test statistic lies within the critical region, we reject the null hypothesis

1. Otherwise we fail to reject it.

1. Test Statistic

• Naturally, we have to sample, then calculateand s.

• From these two values, the sample size, and our hypothesis value, we calculate the test statistic:

• Where μ is our hypothetical benchmark

X

Xz

s

n

For example

• People who eat five servings of fruit and veggies daily have an average life span of 80 years. Our sample of 120 people who has a mean lifespan is 82.5 years (standard deviation is 4.6 years)

• South students have an IQ greater than 115. Our sample of 36 students, their mean IQ is 108 years (standard deviation is 18)

Your turn

• American cars have an average repair cost greater than $600– The mean of a sample of 45 cars was $520 with a standard

deviation of $120.

• The average New Jersey property tax bill is $6,500. – The mean of a sample of 65 homes was $6,700 with a

standard deviation of $3,200.

Homework: Find the test statistic

1. H0 = 110, X-bar = 118, s = 12, n = 45

2. H0 ≥ 6.5, X-bar = 7, s = 1.2, n = 32

3. H0 = 45, X-bar = 50, s = 12, n = 100

4. H0 ≤ 1.1, X-bar = 0.8, s = 0.12, n = 60

5. H0 ≤ 12000, X-bar = 11800, s = 1002, n = 35

6. The average math SAT scores is at least 550– Sample mean is 535, standard deviation = 102, n = 90

7. The average GPA is 3.0– Sample mean is 3.05, standard deviation = 1.02, n = 30

2. Critical Regions

• Area of the standard normal curve outside the degree of confidence– AKA Significance level

– α = 1 – degree of confidence

– Probability that we will reject H0 even though it’s true

• Bounded by the critical value(s)– Will be compared with the test statistic

– Found in the tables

• If the test statistic falls in the critical region we reject H0

Right-tailed Critical Region:

• α (critical region) occupies the right of the curve• Degree of confidence is in the left of the curve

• Usually used when the H0 < benchmark

Left-tailed critical region

• α (critical region) occupies the left of the curve• Degree of confidence is in the right of the curve

• Usually used when the H0 > benchmark

Two-tailed critical region

• Critical region split between the two ends of the curve

• Degree of confidence is in the center of the curve

• Used when H0 is an equality

Critical Values: Large samples

Degree of confidence

Two tailed Left tailed Right tailed

90 1.645 -1.28 1.28

95 1.96 -1.645 1.645

98 2.33 -2.05 2.05

99 2.575 -2.33 2.33

99.9 3.30 -3.10 3.10

For example

1. H0: μ = 801. DoC = 95%

2. H0: μ <= 1151. DoC = 98%

3. H0: μ >= 701. DoC = 99%

Home Work: Find the Critical Values

1. H0: μ = 401. DoC = 98%

2. H0: μ <= 1151. DoC = 90%

3. H0: μ >= 701. DoC = 96%

4. H1: μ > 981. DoC = 95%

5. H1: μ 0.251. DoC = 95%

6. H0: μ = 4.01. DoC = 99.9%

7. H0: μ ≥ 1.151. DoC = 90%

8. H0: μ ≥ 7.01. DoC = 90%

9. H1: μ = 8.91. DoC = 98%

10. H1: μ ≤ 1.251. DoC = 95%

3. Stating our conclusions

• Strictly speaking, we’ve either reject or fail to reject the null hypothesis.

• We sometimes say “accept” instead of “fail to reject”, but this can be misleading, since we really haven’t proven the null hypothesis; only that there’s not enough evidence to disprove it.

What if we’re wrong?

• Type 1 error: Reject a null hypothesis that is true.– False positive– Probability of doing so is α, the significance level

• Type 2 error: Accept a null hypothesis that is false.– False negative– Probability of doing so is β.

• The consequences of these errors will determine degree of confidence and sample size– If α is fixed, increasing n will decrease β– If n is fixed, then decreasing α will increase β (and vice

versa– Increasing n will decrease both α and β

Example

• A vendor claims its batteries can go at least 24 hours before recharging.

• If we pick a 90% degree of confidence…• There’s a 10% probability that we will reject the

claim, even though the actual rate is more that 24 hours

• There’s also a probability we will “accept” the claim even though the rate is less that 24 hours