Today’s Agenda:

Hypothesis Testing on the Mean

Hypothesis Testing on Variance

Hypothesis Testing on Standard Deviation

HypothesisTesting

General Procedure for Hypothesis testing

Determine the Parameter of InterestStep 1:

State the Null HypothesisStep 2:

State the Alternative HypothesisStep 3:

Determine AlphaStep 4:

Determine Test StatisticStep 5:

Determine Rejection RegionStep 6:

Compute Test StatisticStep 7:

ConcludeStep 8:

HypothesisTesting

The similarity of a hypothesis test and a confidence interval

The “acceptance” region of a hypothesis test with a given α is the same as that of the 100(1 – α)% confidence interval

similarly computed given the same data set

More clearly, if the test statistic is outside the computed 100(1 – α)% confidence interval, the hypothesis is rejected

HypothesisTesting

One-sided confidence intervals

A company tests that the burning rate of a certain propellant is less than 50 centimeters per second and wishes to show this

with a strong conclusion

Ho: μ = 50 H1: μ < 50

This satisfies the desired outcome of the experiment

Note that when we fail to reject Ho does not mean that μ = 50, but rather we do not have strong evidence to support H1

HypothesisTesting

Hypothesis Test on Mean, Normal Distribution, Variance Known

n

xz o

o

Hypothesis Test on Mean, Normal Distribution, Variance Unknown

ns

xt oo

HypothesisTesting

Ten measurements of weight of potato chip packs were done for a certain manufacturing company. Historical data states that the

distribution of the weight of potato chip packs are normally distributed with population standard deviation of 1.07g. The weight measurements are found below. Test the hypothesis that the true

mean weight is equal to 101g using a significance level of α = 0.05

Weight measurements of potato chip packs

100.12 99.63 98.55 100.36 100.87

102.04 100.78 101.34 99.97 99.21

HypothesisTesting

Determine the Parameter of InterestStep 1:Ten measurements of weight of potato chip packs were done for a

certain manufacturing company. Historical data states that the distribution of the weight of potato chip packs are normally

distributed with population standard deviation of 1.07g. The weight measurements are found below. Test the hypothesis that the true

mean weight is equal to 101g using a significance level of α = 0.05

Weight measurements of potato chip packs

100.12 99.63 98.55 100.36 100.87

102.04 100.78 101.34 99.97 99.21

We are interested in the mean weight of potato chips μ

HypothesisTesting

State the Null HypothesisStep 2:Ten measurements of weight of potato chip packs were done for a

certain manufacturing company. Historical data states that the distribution of the weight of potato chip packs are normally

distributed with population standard deviation of 1.07g. The weight measurements are found below. Test the hypothesis that the true

mean weight is equal to 101g using a significance level of α = 0.05

Weight measurements of potato chip packs

100.12 99.63 98.55 100.36 100.87

102.04 100.78 101.34 99.97 99.21

Ho: μ = 101

HypothesisTesting

State the Alternative HypothesisStep 3:Ten measurements of weight of potato chip packs were done for a

certain manufacturing company. Historical data states that the distribution of the weight of potato chip packs are normally

distributed with population standard deviation of 1.07g. The weight measurements are found below. Test the hypothesis that the true

mean weight is equal to 101g using a significance level of α = 0.05

Weight measurements of potato chip packs

100.12 99.63 98.55 100.36 100.87

102.04 100.78 101.34 99.97 99.21

H1: μ ≠ 101

HypothesisTesting

Determine AlphaStep 4:Ten measurements of weight of potato chip packs were done for a

certain manufacturing company. Historical data states that the distribution of the weight of potato chip packs are normally

distributed with population standard deviation of 1.07g. The weight measurements are found below. Test the hypothesis that the true

mean weight is equal to 101g using a significance level of α = 0.05

Weight measurements of potato chip packs

100.12 99.63 98.55 100.36 100.87

102.04 100.78 101.34 99.97 99.21

a = 0.05

HypothesisTesting

Determine Test StatisticStep 5:Ten measurements of weight of potato chip packs were done for a

certain manufacturing company. Historical data states that the distribution of the weight of potato chip packs are normally

distributed with population standard deviation of 1.07g. The weight measurements are found below. Test the hypothesis that the true

mean weight is equal to 101g using a significance level of α = 0.05

Weight measurements of potato chip packs

100.12 99.63 98.55 100.36 100.87

102.04 100.78 101.34 99.97 99.21

n

xz o

o

HypothesisTesting

Determine Rejection RegionStep 6:Ten measurements of weight of potato chip packs were done for a

certain manufacturing company. Historical data states that the distribution of the weight of potato chip packs are normally

distributed with population standard deviation of 1.07g. The weight measurements are found below. Test the hypothesis that the true

mean weight is equal to 101g using a significance level of α = 0.05

Weight measurements of potato chip packs

100.12 99.63 98.55 100.36 100.87

102.04 100.78 101.34 99.97 99.21

96.1zz 025.0o

HypothesisTesting

Compute Test StatisticStep 7:Ten measurements of weight of potato chip packs were done for a

certain manufacturing company. Historical data states that the distribution of the weight of potato chip packs are normally

distributed with population standard deviation of 1.07g. The weight measurements are found below. Test the hypothesis that the true

mean weight is equal to 101g using a significance level of α = 0.05

Weight measurements of potato chip packs

100.12 99.63 98.55 100.36 100.87

102.04 100.78 101.34 99.97 99.21

n

xz o

o

1007.1

101287.100 11.2

HypothesisTesting

ConcludeStep 8:Ten measurements of weight of potato chip packs were done for a

certain manufacturing company. Historical data states that the distribution of the weight of potato chip packs are normally

distributed with population standard deviation of 1.07g. The weight measurements are found below. Test the hypothesis that the true

mean weight is equal to 101g using a significance level of α = 0.05

Weight measurements of potato chip packs

100.12 99.63 98.55 100.36 100.87

102.04 100.78 101.34 99.97 99.21

Since the computed zo is greater than the absolute value of 1.96, at 5% level of alpha we reject the

null hypothesisThere is a statistically significant difference on the actual weight of potato chips and hypothesized

mean

HypothesisTesting

Another way to do statistical tests is by using the p-value

HypothesisTesting

P-value

P-value is the smallest level of significance that would lead to rejection of the null hypothesis

))z(1(2p o

HypothesisTesting

Compute for the P-valueTen measurements of weight of potato chip packs were done for a

certain manufacturing company. Historical data states that the distribution of the weight of potato chip packs are normally

distributed with population standard deviation of 1.07g. The weight measurements are found below. Test the hypothesis that the true

mean weight is equal to 101g using a significance level of α = 0.05

Weight measurements of potato chip packs

100.12 99.63 98.55 100.36 100.87

102.04 100.78 101.34 99.97 99.21

Zo= – 2.11

P = 2*(1 – Gaussian(-2.11))

P=0.034858

HypothesisTesting

Compute for the P-value

If p is lowLET IT GO

If the computed p value is lower than the alpha

value, the null hypothesis is rejected

HypothesisTesting

GUIDED EXERCISE 1.1Ten measurements of weight of potato chip packs were done for a

certain manufacturing company. Historical data states that the distribution of the weight of potato chip packs are normally

distributed. The weight measurements are found below. Test the hypothesis that the true mean weight is equal to 101g using a significance level of α = 0.05, and find the p-value of the test

statisticWeight measurements of potato chip

packs

100.12 99.63 98.55 100.36 100.87

102.04 100.78 101.34 99.97 99.21

HypothesisTesting

1. The mean weight of potato chips, μ

2. Ho: μ = 101 g

3. H1: μ ≠ 101 g

4. a = 0.05

5. n

sx

t oo

6. Reject when |to|> t0.025,9 = 2.262

7. 19.2

10027992.1

101287.100

ns

xt oo

8. Since 2.19 < 2.262, we fail to reject Ho at alpha equal to 0.05. There is no significant difference.

HypothesisTesting

GUIDED EXERCISE 1.2

HypothesisTesting

1. The mean coefficient of restitution, μ

2. Ho: μ = 0.82

3. H1: μ > 0.82

4. a = 0.05

5. n

sx

t oo

6. Reject when to > t0.05,14 = 1.761

7. 72.2

1502456.0

82.083725.0

ns

xt oo

8. Since to > 1.761, we reject Ho and conclude at alpha equal to 0.05 the mean coefficient of restitution is greater than 0.82

HypothesisTesting

Hypothesis Test on Variance

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HypothesisTesting

GUIDED EXERCISE 1.3

The sodium content of the manufactured potato chips is assumed to be normally distributed. A random sample of 16

potato chip packs are tested and yielded a variance of 7.5mg2. If the variance of sodium content lacks or exceeds 8mg2, an unacceptable proportion of potato chip packs will not comply

with the set nutritional content. Using α = 0.05, is there statistical evidence in the sample data that suggests that the

manufacturer has a problem with complying with the set nutritional content?

Sourc: Taha

Next Time on IE 27

Hypothesis Testing on Proportion

Computation of Sample Size and Beta

.Fin.