Hypothesis Testing: Examples and Interpretation
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KUS objectives
BAT
Test a hypothesis at a given significance level for
one and two tailed tests
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WB 5
Cars entering a car park have to turn left or right. A survey is done of
20 cars. It is found that 15 went to the left. At the 5% significance level,
does this mean car drivers have a preference for a particular direction?
The evidence supports that we REJECT the hypothesis
There is evidence to suggest that car drivers have a preference for turning left
The probability that 15 or more turned left
Binomial distribution n = 20 x = 15
The evidence supports that we do NOT reject the hypothesis
There is no evidence to support the Doctors claim
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X = number of seeds germinating
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H
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Accept H
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Two tailed tests
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When doing a 2-tail test, the significance level is shared equally
between observations which are less than what was expected and
more than what was expected.
WB 8
Over a long period of time it has been found that in one long
running Mexican restaurant the ratio on non vegetarian to vegetarian
meals is 2:1
In Manuel’s Mexican restaurant in a random sample of 10 people
ordering meals one ordered a vegetarian meal.
Using a 5% level of significance, test whether the proportion of people in
Manuel’s restaurant is different to that of the long running restaurant
Binomial distribution n = 10 x = 1
Do NOT reject the hypothesis
There is no evidence the ratio is different at Manuel’s
WB 9
A manager tells their sales staff that they make a sale to 45% of
customers entering their shop.
The manager randomly selects 40 customers. Of these, 25 were sold
something. At the 5% level of significance, test whether or not what the
manager thinks is justified
Binomial distribution n = 40 x = 35
This is enough evidence to REJECT the hypothesis
There is evidence the staff sell more than the manager says
Practice
Ex 105B
self-assess
One thing learned is –
One thing to improve is –
KUS objectives
BAT
Test a hypothesis at a given significance level for
one and two tailed tests
END
This content covers various examples of hypothesis testing scenarios, including car drivers' preferences for turning directions, the effectiveness of a new drug compared to a standard treatment, and the probability of seeds germinating in a greenhouse. It explains how to formulate null and alternative hypotheses, calculate probabilities, and interpret results at different significance levels. Additionally, it introduces the concept of one-tailed and two-tailed tests in hypothesis testing.
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Presentation Transcript
Hypothesis testing Hypothesis tests
Hypothesis testing BAT Test a hypothesis at a given significance level for one and two tailed tests KUS objectives Starter:
WB 5 Cars entering a car park have to turn left or right. A survey is done of 20 cars. It is found that 15 went to the left. At the 5% significance level, does this mean car drivers have a preference for a particular direction? ?0: ? = 0.5 probability of turning left ?1: ? > 0.5 because we are testing if more turn left (one tailed) Binomial distribution n = 20 x = 15 ?~ ?(20,0.5) The probability that 15 or more turned left P ? 15 = 1 ?(? 14) < 5% = 1 0.9793 = 0.0207 The evidence supports that we REJECT the hypothesis There is evidence to suggest that car drivers have a preference for turning left
The standard treatment for a particular disease has a 2 WB 6 success. A doctor has undertaken research in this area and has produced a new drug which has been successful with 11 out of 20 patients. The doctor claims that the new drug represents an improvement on the standard treatment. a) Write suitable null and alternative hypotheses b) Write the distribution of random variable X that represents the number of patients that could be cured if the null hypothesis were true c) Work out the probability of X taking a value equal to, or greater than 11 d) At a 5% significance level, is this result enough evidence to reject the null hypothesis 5 probability of ?0: ? =2 ?1: ? >2 5 probability of turning left 5 because Doctor claims an improvement (one tailed) 2 5 Binomial distribution ?~? 20, ? ? 11 = 1 ? ? 10 = 1 0.8725 = 0.1275 > 5% The evidence supports that we do NOT reject the hypothesis There is no evidence to support the Doctors claim
WB 7 Brad planted 25 seeds in his greenhouse. He has read in a gardening book that the probability of one of these seeds germinating is 0.25. Ten of Brad s seeds germinated. He claimed that the gardening book had underestimated this probability. Test, at the 5% level of significance, Brad s claim. State your hypotheses clearly. ( ) 25 0 , 25 . ~ X B X = number of seeds germinating H0 : p = 0.25 H1 : p > 0.25 ( ) ( 9287 . ) = = = 10 1 1 9 P X P 0 X 0. 05 0713 0. Accept H0as there is insufficient evidence to support Brad s claim
Two tailed tests So far you have seen what are known as 1-tail tests when H1 is that the parameter in question has either increased or decreased In a 2-tail test, H1 is simply that the parameter in question has changed When doing a 2-tail test, the significance level is shared equally between observations which are less than what was expected and more than what was expected.
WB 8Over a long period of time it has been found that in one long running Mexican restaurant the ratio on non vegetarian to vegetarian meals is 2:1 In Manuel s Mexican restaurant in a random sample of 10 people ordering meals one ordered a vegetarian meal. Using a 5% level of significance, test whether the proportion of people in Manuel s restaurant is different to that of the long running restaurant ?0: ? =1 ?1: ? 1 3 probability vegetarian is ordered 3TWO TAILED ?~ ?( 10,1 Binomial distribution n = 10 x = 1 3 > 2.5% P ? 1 = 0.1040 Do NOT reject the hypothesis There is no evidence the ratio is different at Manuel s
WB 9A manager tells their sales staff that they make a sale to 45% of customers entering their shop. The manager randomly selects 40 customers. Of these, 25 were sold something. At the 5% level of significance, test whether or not what the manager thinks is justified ?0: ? = 0.45 probability of a sale ?1: ? 0.45TWO TAILED ?~ ?( 40,0.45 Binomial distribution n = 40 x = 35 < 2.5% P ? 25 = 1 ? ? 24 = 1 0.9804 = 0.0196 This is enough evidence to REJECT the hypothesis There is evidence the staff sell more than the manager says
Practice Ex 105B
KUS objectives BAT Test a hypothesis at a given significance level for one and two tailed tests self-assess One thing learned is One thing to improve is
