Chapter EightTests of

Hypothesis Based on a

Single Sample

Hypothesis Testing ElementsNull Hypothesis H0:

> Prior BeliefAlternative Hypothesis Ha:

> Contradictory BeliefTest Statistic:

> Parameter used to testRejection Region:

> Set of values for rejecting H0

Hypothesis Testing Errors

Type I Error: > Rejecting H0 when it

is true.

Type II Error: > Not rejecting H0 when

it is false.

Example of Type I ErrorsHighway engineers have found that many factors affect the performance of reflective highway signs. One is the proper alignment of the car’s headlights. It is thought that more than 50% of the cars on the road have misaimed headlights. If this contention can be supported statistically, then a new tougher inspection program will be put into operation. Let p denote the proportion of cars in operation that have misaimed headlights. Setup a test of hypothesis to test this statement.

Type II ErrorsThe reject region for the car test is R = 14, 15, …,20 at = 0.05. Suppose that the true proportion of cars with misaimed headlights is 0.7. What is the probability that our test is unable to detect this situation?

Hypothesis Testing ProtocolIdentify parameter of interest

State Null & Alternative Hypothesis

Give Test Statistic

Find Rejection Region for Level Calculate Sample Size for ()

Decide if H0 is Rejected or Accepted

Hypothesis Test about a Population MeanNormal pdf (known )

Null Hypothesis: H0: u = u0

Test Statistic: z = x – u0

/nAlternative Hypothesis: Reject RegionHa: u > u0 (Upper Tailed) z z

Ha: u < u0 (Lower Tailed) z -z

Ha: u u0 (Two-Tailed)either z z/2

or z -z/2

Hypothesis Testing Mean (Known )Automotive engineers are using more aluminum in the construction of cars in hopes of improving gas mileage. For a particular model the number of miles per gallon obtained currently has a mean of

26.0 mpg with a of 5 mpg. It is hoped that a new design will increase the mean

mileage rating. Assume that is not affected by this change. The sample mean for 49 driving tests with this new design yielded 28.04 mpg. Use a Hypothesis Test

with = .05 to make a decision on the validity of this new design to increase the mean mileage rating.

Hypothesis Testing Mean (Known )As new engineering manager, you test the melting point of 16 samples of hydrogenated vegetable oil from production, resulting in a sample mean of 94.320F. Your company claims a melting point of 950F to all vendors. Using Hypothesis testing, test if your production run meets the 950F specification at level .01. Other evidence indicates a Normal distribution with = 1.20 for the melting point.

Determining ( known)

Alternative Type IIHypothesis Error (u)Ha: u > u0 z+ u0- u

/n

Ha: u < u0 1 - -z+ u0- u /n

Ha: u u0 z/2+ u0- u - -z/2+ u0-

u /n /n

Determining n ( known)

n = (z+ z) 2 One Sided

u0 - u

n = (z/2+ z) 2 Two Sided

u0 - u

Example Type II ErrorAt test level .01, what is the probability of a Type II error when u is actually 940F?

What value of n is necessary to ensure that (94) = .10 when = .01?

Hypothesis Test (Large Sample) Mean

Normal pdf (Unknown )

Null Hypothesis: H0: u = u0

Test Statistic: z = x – u0

s/nAlternative Hypothesis: Reject RegionHa: u > u0 (Upper Tailed) z z

Ha: u < u0 (Lower Tailed) z -z

Ha: u u0 (Two-Tailed)either z z/2

or z -z/2 For & n use plausible values for or use Tables A.17

Hypothesis Test (Large Sample) MeanOzone is a component of smog that can injure sensitive plants even at low levels. In 1979 a federal ozone standard of 0.12 ppm was set. It is thought that the ozone level in air currents over New England exceeds this level. To verify this contention, air samples are obtained from 64 monitoring stations set up across the region. When the data are analyzed, a sample mean of 0.135 and a sample SD of 0.03 are obtained. Use a Hypothesis test at a .01 level of significance to test this theory.

Example HT (Large Sample) MeanThe VP of Sales claims that the salesmen are only averaging 15 sales contacts per week. Looking for ways to increase this figure, the VP selects 49 salesmen at random and the number of contacts is recorded for a week. The sample data reveals a mean of 17 contacts with a sample variance of 9. Does the evidence contradict the VP’s claim at the 5% level of significant?Now the VP wants to detect a difference equal to 1 call in the mean number of customer contacts per week. Specifically, he wants to test u = 15 against u = 16. With the same test data, find for this test.

Hypothesis Test (Small Sample) Mean

Normal pdf (Unknown )

Null Hypothesis: H0: u = u0

Test Statistic: t = x – u0

s/nAlternative Hypothesis: Reject RegionHa: u > u0 (Upper Tailed) t t,v

Ha: u < u0 (Lower Tailed) t -t,v

Ha: u u0 (Two-Tailed)either t t/2,v

or t -t/2,v

To find & Sample Size use Table A.17

Example HT Mean (Small Sample)A new method for measuring phosphorus levels in soil is being tested. A sample of 11 soil specimens with true phosphorus content of 548 mg/kg is analyzed using the new method. The resulting sample mean & sample standard deviation for phosphorus levels are 587 and 10, respectively. Is there evidence that the mean phosphorus level reported by the new method differs significantly from the true value of 548 mg/kg? Use = .05 & assume measurements of this type are Normal.

Example HT Mean (Small Sample)The true average voltage drop from collector to emitter of insulated gate bipolar transistors is supposed to be at most 2.5 volts. A sample of 10 transistors are used to test if the H0: = 2.5 versus Ha: > 2.5 volts

with = .05. If the standard deviation of the voltage distribution is = 0.10, how likely is it that H0

will not be rejected when in fact = 2.6?

Hypothesis Test Population 2

Normal pdf (Unknown )

Null Hypothesis: H0: 2 = 02

Test Statistic: 2 = (n-1)s2

02

Alternative Hypothesis: Reject Region

Ha: 2 > 02 (Upper Tailed) 2 2

,v

Ha: 2 < 02 (Lower Tailed) 2 2

1-,v

Ha: 2 02 (Two-Tailed)either 2 2

/2,v

or 2 21-/2,v

Example HT Variance Indoor swimming pools are noted for their poor acoustical properties. The goal is to design a pool in such a way that the average time it takes a low frequency sound to die is at most 1.3 seconds with a standard deviation of at most 0.6 second. Computer simulations of a preliminary design are conducted to see whether these standards are exceeded. The sample mean was 3.97 seconds and the sample standard deviation was 1.89 seconds for 30 simulations. Does it appear that the design specifications are being met at the = 0.01 level for ?

Example HT VarianceA new process for producing small precision parts is being studied. The process consists of mixing fine metal powder with a plastic binder, injecting the mixture into a mold, and then removing the binder with a solvent.Sample data on parts that should have a 1” diameter and whose standard deviation should not exceed 0.0025 inch yielded a sample mean of 1.00084 with a sample standard deviation of 0.00282 for 15 measured parts. Test at the = .05 level to see if this new process is viable for the population .

P-ValuesSmallest Level ofSignificance at which H0 would be rejected when a specified test procedure is used on a given data set.

Calculating P-Values

One Sided Tests:P = 1 – Φ(z) Upper-tailedP = Φ(z) Lower-tailed

Two Sided Test: P = 2 [ 1 – Φ(|z|) ]