CCT226H5 Lecture Notes - Lecture 9: Null Hypothesis, Statistical Hypothesis Testing, Sampling Distribution
10 May 2020
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Chapter 9: Hypothesis Testing – Tests of Significance
• Developing Null or Alternative Hypotheses
• Population Mean: O Known
• Population Mean: O Unknown
• Population Proportion
Reasoning of significance tests
• We have seen that the properties of the sampling distribution of the sample mean help us
estimate a range of likely values for proportion mean u
o We can also rely on the properties of the sampling distribution to test hypotheses
• Ex. Food company packs tomatoes as 1/2lb (227g). You randomly grab 4 packs and the average
pack is 222g. It’s impossible to get tomatoes to the exact weight, but is the somewhat smaller
weight simply due to chance variation or is it evident that the calibrating machine that sorts
tomatoes in packs needs revision?
• Stating hypotheses
o A test of statistical significance tests a specific hypotheses using sample data to decide
on the validity of the hypothesis
▪ In stats, a hypothesis is an assumption or a theory about the characteristics of
one or more variables in one or more populations
o What you want to know: does the calibrating machine need revision?
▪ The same question statistically: is the population mean u for the distribution of
weights of tomato packages equal to 227g (1/2lb)?
• Hypotheses
o The statement being tested in a test of significance is called the null hypotheses Ho
▪ The test of significance is designed to assess the strength of the evidence
against the null hypotheses
▪ Ho is usually a statement of “no effect” or “no difference”
• In this case, would mean the machine does not need revision and there is
no problem
• Ho: u = 227g
o U is average weight of the population of packs
o The alternative hypotheses is the statement we suspect is true instead of the null
hypothesis
▪ Labeled Ha or H1
• In this case, would mean machine is under-packing packages and needs
revision
o Leads to improvement or change
• Ha: u=/= 227g (u is either larger or smaller)
o Developing null and alternative hypotheses
▪ Testing research hypotheses
• The research hypotheses should be expressed as the alternative
hypotheses
• The conclusion that the research hypothesis is true coms from sample
data that contradict the null hypothesis
o Rejects the null hypothesis by showing a change
▪ Testing the validity of a claim
• Manufacturer’s claims are usually given the benefit of the doubt and
stated as the null hypothesis
• The conclusion that the claim is false comes from sample data that
contradict the null hypothesis
▪ Testing in decision-making situations
• A decision maker might have to choose between two courses of action,
one associated with the null hypothesis and another associated with the
alternative hypothesis
• Ex. accepting a shipment of goods fro a supplier, or returning the
shipment to the supplier
o Summary:
▪ The equality part (=) of the hypotheses always appears in the null hypothesis
▪ In general, a hypothesis test about the value of a population mean u must take
one of the following three forms
• Uo is the hypothesized value of the population mean
• Ho:u>uo, Ha: u<uo
o One-tailed (Lower-tail)
▪ Being lesser than u (ex. lower tomato package weight)
• Ho: u<uo, Ha:u>uo
o One-tailed (Upper-tail)
▪ Being higher than u
• Ho: u=uo, Ha:u=/=uo
o Two-tailed
o Null and Alternative Hypotheses example
▪ Metro EMS has approximately 20 mobile medical units with a goal to respond
with a mean time of 12 minutes or less
• Problem is only on the upper-tail; if they take longer than 12 minutes
(taking less time isn’t a problem)
• Take sample to determine whether or not the service goal is being
achieved
o If Ho: u<12, the emergency service is meeting the response goal
and no follow-up action is necessary
o If Ha: u>12, the emergency service is not meeting the response
goal and appropriate follow-up action is necessary
o Where u=mean response time for the population of medical
emergency requests
• The P (probability) value
o Key number in test
▪ Ex. packaging process for tomatoes has a known
standard deviation of o=5g
• Ho:u=227g vs Ha:u=/=227g, where average
weight from random four boxes is 222g
• What is probability of drawing a sample such as
this is Ho is true?
o Tests of significance quantify the
chance of obtaining a particular random
Document Summary
Chapter 9: hypothesis testing tests of significance: developing null or alternative hypotheses, population mean: o known, population mean: o unknown, population proportion. You randomly grab 4 packs and the average pack is 222g. If ho: u<12, the emergency service is meeting the response goal and no follow-up action is necessary. This quantity is the p-value: this is a way of assessing the. Believability of the null hypothesis given the evidence provided by a random sample. Against ho: how small, when the tail area becomes very small the probability of drawing such a sample at random gets very slim. Highway patrol: one-tailed test about a population mean: o unknown, highway patrol samples vehicle speeds at various locations. Sample speed is used to test hypothesis ho: u<65: locations where ho is rejected are deemed the best locations for radar traps. Location f, sample of 64 vehicles shows mean speed of 66. 2 mph with sd of.
