section 10.1

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Section 10.1

Fundamentals of Hypothesis Testing

With more helpful content added by D.R.S., University of Cordele

Hypothesis testing is a technique for testing a claim about a population parameter using statistical principles.






Null and Alternative Hypotheses

Null and Alternative HypothesesThe alternative hypothesis, denoted by is a mathematical statement that describes a population parameter, and it is the hypothesis that the researcher is aiming to gather evidence in favor of; it is also referred to as the research hypothesis.

,aH

The subscript is a lowercase little letter “a”






Null and Alternative Hypotheses

Null and Alternative Hypotheses (cont.)The null hypothesis, denoted by is the mathematical opposite of the alternative hypothesis; it will always include equality.

0 ,H

The subscript is the digit zero, 0.

This means the Null Hypothesis willalways involve the symbol “=“ or “≥” or “≤”.

And the Alternative Hypothesis willalways involve the symbol “≠” or “<“ or “>”.






What “population parameter” is it?

Sometimes it’s a hypothesis test about μ, which is the _____________________ _________value of some measurement.Sometimes it’s a hypothesis test about p,the ________________ of the population that has some characteristic (a certain medical condition, or owns some product, or prefers some flavor, etc.)

Hypothesis testing is a technique for testing a claim about a population parameter using statistical principles.






Example of Writing Hypotheses

An engineer has designed a valve… tested on 180 engines and the mean pressure was 4.8 lbs/sq.in. Assume σ is known; σ =1. If the valve was designed to produce a mean pressure of 4.6 lbs/sq.in., is there sufficient evidence at the 0.05 level that the value does not perform to specifications?

H0: ____________________ (the default “fact”)

Ha: ____________________ (the contrary claim)







A manufacturer of chocolate chips would like to know whether its bag filling machine works correctly at the 440 gram setting. Is there sufficient evidence at the 0.02 level that the bags are overfilled? Assume the population is normally distributed.



Sometimes it’s a good idea to start with the

Alternative Hypothesis, then go back to

the Null Hypothesis.







Using traditional methods it takes 106 hours to receive an advanced license. A new and different training program has been proposed. A researcher believes the new technique may lengthen the training time and examined a sample of 60 students…



Do I write hypotheses using μ (mu) or p ?

If the hypotheses are about a population mean, use μ.

If the hypotheses are about a population proportion use p.

If the hypotheses are about a population percentage, it is really a proportion problem, so p is the right letter.







A sample of 900 computer chips revealed that 4.1% of the chips fail in the first 1000 hours of use. The company’s promotional literature says that 4.5% fail in the first 1000 hours of use. Is there sufficient evidence at the 0.02 level to disprove the claim?









Company claims that 99% of its customers are satisfied. A competitor thinks that’s inflated and commissions a secret survey of a random sample of 77 of those customers.



Sometimes it’s a good idea to start with the

Alternative Hypothesis, then go back to

the Null Hypothesis.







A troublemaking busybody excuse of a newspaper published a story saying that only 5 out of 8 graduates of Tick Tock Tech find jobs related to their major field within a year of graduation. T.T.T. takes a random sample of 135 graduates and learns that 93 of them have successful, rewarding, degree-related jobs.



Where does the “=” live?

The Null Hypothesis, H0, always has the “=” sense: = or ≤ or ≥.

The Alternative Hypothesis, Ha, always has a strict inequality: ≠ or < or >

The possibilities are:

• H0:μ = some value and Ha:μ ≠ that value

• H0:μ ≤ some value and Ha:μ > that value

• H0:μ ≥ some value and Ha:μ < that value

And similarly for proportions with p.






What is a Test Statistic ? And what does “Statistically Significant” mean?

Test StatisticA test statistic is the value used to make a decision about the null hypothesis and is derived from the sample statistic.A sample statistic is said to be statistically significant if it is far enough away from the presumed value of the population parameter to conclude that it would be unlikely for the sample statistic to occur by chance if the null hypothesis is true.






Test Statistic and the Level of Significance

Test Statistic (cont.)The level of significance, denoted by a, is the probability of making the error of rejecting a true null hypothesis in a hypothesis test; a = 1 - c.

The Level of Significance, α (alpha), represents the small area in the tail(s), That’s where Ha “wins”, because we reject H0.

The confidence level, c, represents the big area in which Ha “loses”,because we fail to reject H0,for lack of strong enough evidence.






Test Statistic

Conclusions for a Hypothesis Test • Reject the null hypothesis. • Fail to reject the null hypothesis.

Your hypothesis test will always conclude with one of these two endings.• Either you have strong enough evidence to reject H0.• Or you don’t have strong enough evidence.

• We do NOT say we “accept Ha” or “accept H0”.• Rather, we say we “fail to reject H0”.






Interpreting the Conclusion to a Hypothesis Test

A city official claimed that stationing an officer outside the elementary school just one morning a week was enough to slow the average driver to 28 mph. Some “concerned” parent said the cars were really traveling much faster than that and demanded more action.

A group of Criminal Justice students ran a radar speed check on 81 randomly selected vehicles. And some statistics students designed and performed a hypothesis test. Their decision was “fail to reject the null hypothesis.”







The hypotheses are:H0: _______________ and Ha: ________________

The conclusion is(a) There is sufficient evidence at the _____ level of

significance that drivers are still speeding.(b) There is not sufficient evidence at the _____ level of

significance that drivers are still speeding.







Motorhead Monthly Magazine believes that 50% of their readership plans to buy a brand new vehicle within the next twelve months. They hire a consulting firm that • decides to use a 0.02 level of significance, • chooses a random sample of readers, • crunches the data, • calculates the test statistic, • and determines that the correct decision is to reject

the null hypothesis.







The hypotheses are:H0: _______________ and Ha: ________________

The conclusion is(a) There is sufficient evidence at the 0.02 level of

significance that the percentage is not 50%.(b) There is not sufficient evidence at the 0.02 level of

significance that the percentage is not 50%.






How do you do a Hypothesis Test?

Performing a Hypothesis Test 1. State the null and alternative hypotheses. 2. Determine which distribution to use (z or t) for the test statistic, and state the level of significance (decide what is the value of alpha, α)3. Gather data and calculate the necessary sample

statistics. 4. Draw a conclusion (reject H0 or fail to reject H0) and interpret the decision.






• Type I Error – the probability of rejecting a true null hypothesis, a.

• Type II Error – the probability of failing to reject a false null hypothesis , b.

Definitions:

Types of Errors

The RealityH0 is true H0 is false

Your Decision

H0 is rejected Type I error Correct Decision

H0 is not rejected Correct Decision Type II error

Does your Decision agree with Reality ?

Consider: what might lead to an error? Who or What is at fault?






Example 10.7: Determining the Type of Error

A television executive believes that at least 99% of households in the United States have at least one television. An intern at the executive’s company is given the task of using a hypothesis test to determine whether the percentage is actually less than 99%. The hypothesis test is completed, and based on the sample collected, the intern decides to fail to reject the null hypothesis. If, in reality, 96.7% of households own a television set, was an error made? If so, what type?






Example 10.7: Determining the Type of Error (cont.)

Write hypotheses. Think: “What is the intern gathering data for?” That decides Ha, and H0 follows.

H0: __________ Ha: __________


Your Decision



Which box represents what happened?

based on the sample collected, the intern decides to fail to reject the null hypothesis. Suppose that, in reality, 96.7% of households own a television set.







Insurance companies commonly use 1000 miles as the mean number of miles a car is driven per month. One insurance company claims that, due to our more mobile society, the mean is more than 1000 miles per month. The insurance company tests its claim with a hypothesis test and decides to reject the null hypothesis. Assume that in reality, the mean number of miles a car is driven per month is 1250 miles. Was an error made? If so, what type?






Example 10.8: Determining the Type of Error (cont.)

The decision was to { reject, fail to reject } the null hypothesis. So what happened?


Your Decision



Hypotheses: H0: ___________ Ha: _____________







A study on the effects of television-viewing on children reports that children watch a mean of 4.0 hours of television per night. Kiko believes that the mean number of hours children in her neighborhood watch television per night is not 4.0. She performs a hypothesis test and rejects the null hypothesis. Assume that in reality, children in her neighborhood do watch a mean of 4.0 hours of television per night. Did she make an error? If so, what type?






Example 10.9 Kiko’s TV viewing test

The decision was to { reject, fail to reject } the null hypothesis. So what happened?


Your Decision



Hypotheses: H0: ___________ Ha: _____________

section 10.1

Documents

research hypothesis

null hypothesis willalways

success counts

alternative hypothesesnull

population parameter

alternative hypotheses

sufficient evidence

section x