Math 1530 Elements of Statistics

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Math 1530 Elements of Statistics

• Chapter 9
• Hypothesis Tests for One Population Mean

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Example

• Suppose a company produces a bags of pretzels,
  claiming that the average mass per bag is 454 g
  with a standard deviation of 7.8 g.
• The quality assurance guys samples 25 random bags
  and finds an average mass of 450g.
• Is there any reason to be concerned?
• Use a method called hypothesis testing to answer
  the question above.

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Definitions

• A hypothesis is an assumption which may or may
  not be true.
• In our example, our hypothesis would be the
  average mass of bag of pretzel is 454 g.

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Definitions

• Null hypothesis
• A hypothesis to be tested
• denoted by H0
• When testing hypothesis about population means,
  the null hypothesis takes the form
• H0 µ µ0 where µ0 is some number
• For our example, the null hypothesis would be
• H0 µ 454 g

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Definitions

• Alternative hypothesis
• A hypothesis considered as an alternative to the
  null hypothesis.
• It is a new idea.
• denoted by Ha
• When testing hypothesis about population means,
  the alternative hypothesis takes one of the three
  forms.
• Two-tailed hypothesis test The population mean
  is different from a specific value.
• Left-tailed hypothesis test The population mean
  is less than a specific value.
• Right-tailed hypothesis test The population mean
  is greater than a specific value.
• A hypothesis test is called a one-tailed test if
  it is either left tailed or right tailed.

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Examples

• Pg 385
• Q 9.6
• Q 9.8
• Q 9.10

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Basic Logic of Hypothesis Testing

• Take a random sample from the population.
• If sample data are consistent with the null
  hypothesis, do not reject H0.
• If sample data are inconsistent with H0 (in the
  direction of the alternative hypothesis), reject
  H0 in favor of Ha.

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Example Pretzels

• A company uses a machine to produce bags of
  pretzels with mean mass 454 g. Assume that the
  net weight of bags is normally distributed. The
  Quality Assurance guys would like to test the
  working of the machine. They randomly pick 25
  bags whose masses are shown in the table. Does
  the data provide sufficient evidence to conclude
  that the machine is not working properly?
• Assume the masses of the pretzel bags have a
  normal distribution with standard deviation 7.8
  g.

Taken from Elements of Statistics, by Neil Weiss

• State H0 and Ha.
• Discuss the logic behind the test.
• What is the distribution of for samples of
  size 25?

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Criterion for rejecting H0 in favor of Ha

• If the mean weight, , of the 25 bags of
  pretzels sampled is more than two standard
  deviations (3.12 g) from 454 g, reject the null
  hypothesis and conclude that the alternative
  hypothesis is true. Otherwise, do not reject the
  null hypothesis.

Taken from Elements of Statistics, by Neil Weiss
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Apply this Criterion to the sample data

• The mean weight, , of the sample of 25 bags of
  pretzels whose weights are given is 450g.
• So, z ( - 454) / 1.56 (450 - 454) /1.56
  -2.56.
• Thus, the sample mean of 450 g is 2.56 standard
  deviations below the null-hypothesis population
  mean of 454 g, as shown in the figure

Taken from Elements of Statistics, by Neil Weiss
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Terminology for Hypothesis Tests

• The test statistic is the statistic used for
  deciding whether or not the null hypothesis
  should be rejected.
• Rejection region is the set of values for the
  test statistic that leads to the rejection of the
  null hypothesis.
• Non-Rejection region is the set of values for the
  test statistic that leads to the non-rejection of
  the null hypothesis.
• Critical values are the numbers that separate the
  rejection and non-rejection regions.

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Rejection Regions and Critical Values
Taken from Elements of Statistics, by Neil Weiss
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Type I and Type II Error

• The significance level is the probability of
  making a Type I error.
• is the probability of making Type II error,
  and depends on the true value of .
• When performing hypothesis tests, we generally
  want to minimize the error probabilities and
  .

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Power of a hypothesis Test

• The power of a hypothesis test is the probability
  of not making a Type II error,
• Thus,
• Power 1- P( Type II error)
• 1- ß

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Taken from Elements of Statistics, by Neil Weiss
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Example ( Agricultural Books)

• Pg 393 Q 9.30
• In 2000, the mean retail price of agriculture
  books was 66.52. A hypothesis test is to be
  performed to decide whether this years main
  retail price of agriculture books has changed
  from the 2000 mean. The null and the alternative
  hypothesis are
• H0 µ 66.52 and
• Ha µ ? 66.52
• where µ is this years mean retail price of
  agricultural books. Explain what each of the
  following would mean.
• Type 1 error Type II error Correct Decision

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Example ( Agricultural Books)

• Pg 393 Q 9.30
• Now suppose that the results of carrying out the
  hypothesis test leads to the rejection of the
  null hypothesis. Classify that conclusion by
  error type or as a correct decision if in fact
  this years retail price of a agriculture books.
• Equals the 2000 mean of 66.52
• Differs from the 2000 mean of 66.52

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Example Early-Onset Dementia

• Pg 394 Q 9.32
• A hypothesis test is to be performed to decide
  whether the
• mean age at diagnosis of all people with
  early-onset dementia
• is less than 55 years old. The null and the
  alternative
• hypothesis are
• H0 µ 55 years old and
• Ha µ lt 55 years old
• where µ is the mean age at diagnosis of all
  people with early onset dementia. Explain what
  each of the following would mean.
• Type 1 error Type II error Correct Decision

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Example Early-Onset Dementia

• Pg 394 Q 9.32
• Now suppose that the results of carrying out the
  hypothesis test leads to the nonrejection of the
  null hypothesis. Classify that conclusion by
  error type or as a correct decision if in fact
  the mean age at diagnosis of all people with
  early onset dementia
• is 55 years old.
• is less than 55 years old.

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Obtaining Critical Values

• Suppose that a hypothesis test is to be performed
  at the significance level, a, then the critical
  values must be chosen so that, if the null
  hypothesis is true, the probability is a that the
  test statistic will fall in the rejection region.

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Examples Critical Values

• Pg 405
• A hypothesis test is to be performed for a
  population mean with a null hypothesis of H0 µ
  µ0. Further suppose that the test statistic is
  ,
• Determine the critical values for the following
  and sketch the graph
• (Q 9.44) A two-tailed test with a 0.10
• (Q 9.46) A left-tailed test with a 0.01
• (Q 9.48) A right-tailed test with a 0.01

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One Sample z-Test for a Population Mean

• Necessary assumptions
• 1. Simple random sample.
• 2. Population has a normal distribution,
• or the sample size is large (n 30).
• 3. s is known.
• State H0 and Ha.
• Set the significance level a and determine the
  critical value(s).
• Compute the test statistic
• Reject H0 in favor of Ha if the test statistic is
  in the rejection region, otherwise do not reject
  H0.

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Example Probability Books

• The mean retail price of probability books was
  81 in 2002.
• 35 random retail prices of probability books (in
  ) from this year are
• At the 1 significance level, do the data provide
  sufficient evidence that this years mean price
  increased? Assume s 7.

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P-Values

• Assuming H0 is indeed true, the P-value is the
  probability of observing a value of the test
  statistic as or more extreme than that observed.
• Small P-values (close to 0) provide evidence
  against the null hypothesis.
• Large p-values do not.

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P-Values

• For a right-tailed test,
• P-value P(Z gt z0).
• For a two-tailed test,
• P-value P(Z gtz0).
• For a left-tailed test,
• P-value P(Z lt z0).

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Examples

• Pg 416
• Q. 9.90
• Q. 9.92
• Q. 9.94

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Decision Criteria for a hypothesis test using the
P-Value

• If the P-value is less than or equal to
  significance level, reject the null hypothesis (
  H0 ) otherwise we do not.
• Example ( Pg 416)
• Q. 9.78

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Guidelines for using P-values
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One Sample z-Test for a Population Mean The
P-value Approach

• Necessary assumptions
• 1. Simple random sample.
• 2. Population has a normal distribution,
• or the sample size is large (n 30).
• 3. s is known.
• State H0 and Ha.
• Set the significance level a.
• Compute the test statistic
• Reject H0 if P-value lt a.

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Example Calcium from Elementary Statistics by
Neil Weiss

• Daily calcium intakes (in mg) for 18 randomly
  selected people below the poverty level are
• Assume the population is approximately normal and
  s 188 mg.
• Do the data provide sufficient evidence to
  conclude at the 5 significance level the mean
  calcium intake of people below the poverty level
  is less that the RDA of 800 mg?

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Hypothesis Test for One Population Mean When s
is unknown

• When the population standard deviation is
  unknown, we use the sample standard deviation.
• Use t-values instead of the z-values
• Assume a simple random sample and a normal
  population or large sample with n 30.
• Use either the critical value or the P-value
  approach

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P-Values for the t-test

• For a right-tailed test,
• P-value P(t gt t0).
• For a two-tailed
  test,
• P-value P(t
  gtt0).
• For a left-tailed test,
• P-value P(t lt t0).

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One Sample t-Test for a Population Mean

• Necessary assumptions
• 1. Simple random sample.
• 2. Population has a normal distribution,
• or the sample size is large (n 30).
• 3. s is unknown.
• State H0 and Ha.
• Set the significance level a and determine the
  critical value(s).
• Compute the test statistic
• Reject H0 in favor of Ha if t is in the rejection
  region, and fail to reject H0 otherwise.

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Examples

• Pg 429
• Q. 9.124

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Hypothesis Tests Summary

• Two kinds of hypothesis tests for population
  means
• When s is known, perform a z-test.
• When s is not known, perform a t-test.
• Null hypotheses for population means take form of
  µ µ0.
• Alternative hypotheses for population means take
  one of three forms
• Can also test using the P-value approach.

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Finish reading Chapter 9 and do the homework.