 # Chapter 10 Section 2 Hypothesis Tests for a Population Mean

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Chapter 10 Section 2 Hypothesis Tests for a Population Mean
Assuming the Population Standard Deviation is Known

Chapter 10 – Section 2 Learning objectives
Understand the logic of hypothesis testing Test hypotheses about a population mean with σ known using the classical approach Test hypotheses about a population mean with σ known using P-values Test hypotheses about a population mean with σ known using confidence intervals Understand the difference between statistical significance and practical significance 1 2 3 5 4

Chapter 10 – Section 2 Learning objectives
Understand the logic of hypothesis testing Test hypotheses about a population mean with σ known using the classical approach Test hypotheses about a population mean with σ known using P-values Test hypotheses about a population mean with σ known using confidence intervals Understand the difference between statistical significance and practical significance 1 2 3 5 4

Chapter 10 – Section 2 We have the outline of a hypothesis test, just not the detailed implementation How do we quantify “unlikely”? How do we calculate Type I and Type II errors? What is the exact procedure to get to a do not reject / reject conclusion?

Chapter 10 – Section 2 There are three equivalent ways to perform a hypothesis test They will reach the same conclusion The methods The classical approach The P-value approach The confidence interval approach

Chapter 10 – Section 2 The classical approach The P-value approach
If the sample value is too many standard deviations away, then it must be too unlikely The P-value approach If the probability of the sample value being that far away is small, then it must be too unlikely The confidence interval approach If we are not sufficiently confident that the parameter is likely enough, then it must be too unlikely Don’t worry … we’ll be explaining more The classical approach If the sample value is too many standard deviations away, then it must be too unlikely The P-value approach If the probability of the sample value being that far away is small, then it must be too unlikely The confidence interval approach If we are not sufficiently confident that the parameter is likely enough, then it must be too unlikely The classical approach If the sample value is too many standard deviations away, then it must be too unlikely The P-value approach If the probability of the sample value being that far away is small, then it must be too unlikely The classical approach If the sample value is too many standard deviations away, then it must be too unlikely

Chapter 10 – Section 2 The three methods all begin the same way
We have a null hypothesis, that the actual mean is equal to a value μ0 We have an alternative hypothesis The three methods all set up a criterion A criterion that quantifies “unlikely” That the actual mean is unlikely to be equal to μ0 A criterion that determines what would be a do not reject and what would be a reject The three methods all begin the same way We have a null hypothesis, that the actual mean is equal to a value μ0 We have an alternative hypothesis

Chapter 10 – Section 2 The three methods all need information
We run an experiment We collect the data We calculate the sample mean The three methods all make the same assumptions to be able to make the statistical calculations That the sample is a simple random sample That the sample mean has a normal distribution The three methods all need information We run an experiment We collect the data We calculate the sample mean

Chapter 10 – Section 2 In this section we assume that the population mean σ is known (as in section 9.1) We can apply our techniques if either The population has a normal distribution Our sample size n is large (n ≥ 30) In those cases, the distribution of the sample mean is normal with mean μ and standard deviation σ / √ n

Chapter 10 – Section 2 The three methods all compare the observed results with the previous criterion Classical – how many standard deviations P-value – the size of the probability Confidence interval – inside or outside the interval If the results are unlikely based on these criterion, then we say that the result is statistically significant

Chapter 10 – Section 2 The three methods all conclude similarly
We do not reject the null hypothesis, or We reject the null hypothesis We reject the null hypothesis when the result is statistically significant

Chapter 10 – Section 2 We now will cover how each of the
Classical P-value, and Confidence interval approaches will show us how to conclude whether the result is statistically significant or not

Chapter 10 – Section 2 Learning objectives
Understand the logic of hypothesis testing Test hypotheses about a population mean with σ known using the classical approach Test hypotheses about a population mean with σ known using P-values Test hypotheses about a population mean with σ known using confidence intervals Understand the difference between statistical significance and practical significance 1 2 3 5 4

Chapter 10 – Section 2 The classical approach
We compare the sample mean to the hypothesized population mean μ0 Measure the difference in units of standard deviations A lot of standard deviations is far … few standard deviations is not far Just like using a general normal distribution The classical approach We compare the sample mean to the hypothesized population mean μ0 Measure the difference in units of standard deviations A lot of standard deviations is far … few standard deviations is not far The classical approach We compare the sample mean to the hypothesized population mean μ0 Measure the difference in units of standard deviations The classical approach We compare the sample mean to the hypothesized population mean μ0

Chapter 10 – Section 2 How far is too far?
For example, we can use α = 0.05 as the level of significance “Unlikely” means that this difference occurs with probability α = 0.05 of the time, or less This concept applies to two-tailed tests, left-tailed tests, and right-tailed tests How far is too far? For example, we can use α = 0.05 as the level of significance “Unlikely” means that this difference occurs with probability α = 0.05 of the time, or less How far is too far? For example, we can use α = 0.05 as the level of significance How far is too far?

Chapter 10 – Section 2 For two-tailed tests For two-tailed tests
The least likely 5% is the lowest 2.5% and highest 2.5% (below –1.96 and above standard deviations) … –1.96 and are the critical values The region outside this is the rejection region For two-tailed tests The least likely 5% is the lowest 2.5% and highest 2.5% (below –1.96 and above standard deviations) … –1.96 and are the critical values

Chapter 10 – Section 2 For left-tailed tests For left-tailed tests
The least likely 5% is the lowest 5% (below –1.645 standard deviations) … –1.645 is the critical value The region less than this is the rejection region For left-tailed tests The least likely 5% is the lowest 5% (below –1.645 standard deviations) … –1.645 is the critical value

Chapter 10 – Section 2 For right-tailed tests For right-tailed tests
The least likely 5% is the highest 5% (above standard deviations) … is the critical value The region greater than this is the rejection region For right-tailed tests The least likely 5% is the highest 5% (above standard deviations) … is the critical value

Chapter 10 – Section 2 An example of a two-tailed test
A bolt manufacturer claims that the diameter of the bolts average 10.0 mm H0: Diameter = 10.0 H1: Diameter ≠ 10.0 We take a sample of size 40 (Somehow) We know that the standard deviation of the population is 0.3 mm The sample mean is mm We’ll use a level of significance α = 0.05 An example of a two-tailed test An example of a two-tailed test A bolt manufacturer claims that the diameter of the bolts average 10.0 mm H0: Diameter = 10.0 H1: Diameter ≠ 10.0

Chapter 10 – Section 2 Do we reject the null hypothesis?
10.12 is 0.12 higher than 10.0 The standard error is (0.3 / √ 40) = 0.047 The test statistic is 2.53 The critical normal value, for α/2 = 0.025, is 1.96 2.53 is more than 1.96 Our conclusion We reject the null hypothesis We have sufficient evidence that the population mean diameter is not 10.0 Do we reject the null hypothesis? 10.12 is 0.12 higher than 10.0 The standard error is (0.3 / √ 40) = 0.047 The test statistic is 2.53 The critical normal value, for α/2 = 0.025, is 1.96 2.53 is more than 1.96

Chapter 10 – Section 2 An example of a left-tailed test
A car manufacturer claims that the mpg of a certain model car is at least 29.0 H0: MPG = 29.0 H1: MPG < 29.0 We take a sample of size 40 (Somehow) We know that the standard deviation of the population is 0.5 The sample mean mpg is 28.89 We’ll use a level of significance α = 0.05 An example of a left-tailed test An example of a left-tailed test A car manufacturer claims that the mpg of a certain model car is at least 29.0 H0: MPG = 29.0 H1: MPG < 29.0

Chapter 10 – Section 2 Do we reject the null hypothesis?
28.89 is 0.11 lower than 29.0 The standard error is (0.5 / √ 40) = 0.079 The test statistic is -1.39 -1.39 is greater than , the left-tailed critical value for α = 0.05 Our conclusion We do not reject the null hypothesis We have insufficient evidence that the population mean mpg is less than 29.0 Do we reject the null hypothesis? 28.89 is 0.11 lower than 29.0 The standard error is (0.5 / √ 40) = 0.079 The test statistic is -1.39 -1.39 is greater than , the left-tailed critical value for α = 0.05

Chapter 10 – Section 2 An example of a right-tailed test
A bolt manufacturer claims that the defective rate of their product is at most 1.70 per 1,000 H0: Defect Rate = 1.70 H1: Defect Rate > 1.70 We take a sample of size 40 (Somehow) We know that the standard deviation of the population is .06 The sample defect rate is 1.78 We’ll use a level of significance α = 0.05 An example of a right-tailed test An example of a right-tailed test A bolt manufacturer claims that the defective rate of their product is at most 1.70 per 1,000 H0: Defect Rate = 1.70 H1: Defect Rate > 1.70

Chapter 10 – Section 2 Do we reject the null hypothesis?
1.78 is 0.08 higher than 1.70 The standard error is (0.06 / √ 40) = 0.009 The test statistic is 8.43 8.43 is more than 1.645, the right-tailed critical value for α = 0.05 Our conclusion We reject the null hypothesis We have sufficient evidence that the population mean rate is more than 1.70 Do we reject the null hypothesis? 1.78 is 0.08 higher than 1.70 The standard error is (0.06 / √ 40) = 0.009 The test statistic is 8.43 8.43 is more than 1.645, the right-tailed critical value for α = 0.05

Chapter 10 – Section 2 Two-tailed test Left-tailed test
The critical values are zα/2 and –zα/2 The rejection region is {less than –zα/2} and {greater than z1-α/2} Left-tailed test The critical value is –zα The rejection region is {less than –zα} Right-tailed test The critical value is zα The rejection region is {greater than zα} Two-tailed test The critical values are zα/2 and –zα/2 The rejection region is {less than –zα/2} and {greater than z1-α/2} Two-tailed test The critical values are zα/2 and –zα/2 The rejection region is {less than –zα/2} and {greater than z1-α/2} Left-tailed test The critical value is –zα The rejection region is {less than –zα}

Chapter 10 – Section 2 The difference is
In units of standard deviations, this is This is called the test statistic If the test statistic is in the rejection region – we reject

Chapter 10 – Section 2 The general picture for a level of significance α

Chapter 10 – Section 2 Learning objectives
Understand the logic of hypothesis testing Test hypotheses about a population mean with σ known using the classical approach Test hypotheses about a population mean with σ known using P-values Test hypotheses about a population mean with σ known using confidence intervals Understand the difference between statistical significance and practical significance 1 2 3 5 4

Chapter 10 – Section 2 The P-value is the probability of observing a sample mean that is as or more extreme than the observed The probability is calculated assuming that the null hypothesis is true We use the P-value to quantify how unlikely the sample mean is

Chapter 10 – Section 2 Just like in the classical approach, we calculate the test statistic We then calculate the P-value, the probability that the sample mean would be this, or more extreme, if the null hypothesis was true The two-tailed, left-tailed, and right-tailed calculations are slightly different Just like in the classical approach, we calculate the test statistic Just like in the classical approach, we calculate the test statistic We then calculate the P-value, the probability that the sample mean would be this, or more extreme, if the null hypothesis was true

Chapter 10 – Section 2 For the two-tailed test, the “unlikely” region are values that are too high and too low Small P-values corresponds to situations where it is unlikely to be this far away For the two-tailed test, the “unlikely” region are values that are too high and too low

Chapter 10 – Section 2 For the left-tailed test, the “unlikely” region are values that are too low Small P-values corresponds to situations where it is unlikely to be this low For the left-tailed test, the “unlikely” region are values that are too low

Chapter 10 – Section 2 For the right-tailed test, the “unlikely” region are values that are too high Small P-values corresponds to situations where it is unlikely to be this high For the right-tailed test, the “unlikely” region are values that are too high

Chapter 10 – Section 2 For all three models (two-tailed, left-tailed, right-tailed) The larger P-values mean that the difference is not relatively large … that it’s not an unlikely event The smaller P-values mean that the difference is relatively large … that it’s an unlikely event

Chapter 10 – Section 2 Larger P-values Smaller P-values
A P-value of 0.30, for example, means that this value, or more extreme, could happen 30% of the time 30% of the time is not unusual Smaller P-values A P-value of 0.01, for example, means that this value, or more extreme, could happen only 1% of the time 1% of the time is unusual Larger P-values A P-value of 0.30, for example, means that this value, or more extreme, could happen 30% of the time 30% of the time is not unusual

Chapter 10 – Section 2 The decision rule is For a significance level α
Do not reject the null hypothesis if the P-value is greater than α Reject the null hypothesis if the P-value is less than α For example, if α = 0.05 A P-value of 0.30 is likely enough, compared to a criterion of 0.05 A P-value of 0.01 is unlikely, compared to a criterion of 0.05

Chapter 10 – Section 2 An example of a two-tailed test
A bolt manufacturer claims that the diameter of the bolts average 10.0 mm H0: Diameter = 10.0 H1: Diameter ≠ 10.0 We take a sample of size 40 (Somehow) We know that the standard deviation of the population is 0.3 mm The sample mean is mm We’ll use a level of significance α = 0.05 An example of a two-tailed test An example of a two-tailed test A bolt manufacturer claims that the diameter of the bolts average 10.0 mm H0: Diameter = 10.0 H1: Diameter ≠ 10.0

Chapter 10 – Section 2 Do we reject the null hypothesis?
10.12 is 0.12 higher than 10.0 The standard error is (0.3 / √ 40) = 0.047 The test statistic is 2.53 The 2-sided P-value of 2.53 is 0.01 < 0.05 = α Our conclusion We reject the null hypothesis We have sufficient evidence that the population mean diameter is not 10.0 Do we reject the null hypothesis? 10.12 is 0.12 higher than 10.0 The standard error is (0.3 / √ 40) = 0.047 The test statistic is 2.53 The 2-sided P-value of 2.53 is 0.01 < 0.05 = α

Chapter 10 – Section 2 An example of a left-tailed test
A car manufacturer claims that the mpg of a certain model car is at least 29.0 H0: MPG = 29.0 H1: MPG < 29.0 We take a sample of size 40 (Somehow) We know that the standard deviation of the population is 0.5 The sample mean mpg is 28.89 We’ll use a level of significance α = 0.05 An example of a left-tailed test An example of a left-tailed test A car manufacturer claims that the mpg of a certain model car is at least 29.0 H0: MPG = 29.0 H1: MPG < 29.0

Chapter 10 – Section 2 Do we reject the null hypothesis?
28.89 is 0.11 lower than 29.0 The standard error is (0.5 / √ 40) = 0.079 The test statistic is -1.39 The 1-sided P-value of is 0.08 > 0.05 = α Our conclusion We do not reject the null hypothesis We have insufficient evidence that the population mean mpg is less than 29.0 Do we reject the null hypothesis? 28.89 is 0.11 lower than 29.0 The standard error is (0.5 / √ 40) = 0.079 The test statistic is -1.39 The 1-sided P-value of is 0.08 > 0.05 = α

Chapter 10 – Section 2 An example of a right-tailed test
A bolt manufacturer claims that the defective rate of their product is at most 1.70 per 1,000 H0: Defect Rate = 1.70 H1: Defect Rate > 1.70 We take a sample of size 40 (Somehow) We know that the standard deviation of the population is .06 The sample defect rate is 1.78 We’ll use a level of significance α = 0.05 An example of a right-tailed test An example of a right-tailed test A bolt manufacturer claims that the defective rate of their product is at most 1.70 per 1,000 H0: Defect Rate = 1.70 H1: Defect Rate > 1.70

Chapter 10 – Section 2 Do we reject the null hypothesis?
1.78 is 0.08 higher than 1.70 The standard error is (0.06 / √ 40) = 0.009 The test statistic is 8.43 The 1-sided P-value of 8.43 is extremely small Our conclusion We reject the null hypothesis We have sufficient evidence that the population mean rate is more than 1.70 Do we reject the null hypothesis? 1.78 is 0.08 higher than 1.70 The standard error is (0.06 / √ 40) = 0.009 The test statistic is 8.43 The 1-sided P-value of 8.43 is extremely small

Chapter 10 – Section 2 Compare the rejection regions for the classical approach and the P-value approach They are the same Classical P-Value

Chapter 10 – Section 2 Learning objectives
Understand the logic of hypothesis testing Test hypotheses about a population mean with σ known using the classical approach Test hypotheses about a population mean with σ known using P-values Test hypotheses about a population mean with σ known using confidence intervals Understand the difference between statistical significance and practical significance 1 2 3 5 4

Chapter 10 – Section 2 The confidence interval approach yields the same result as the classical approach and as the P-value approach We compare A hypothesis test with a level of significance α to A confidence interval with confidence (1 – α) •100% These are the same α’s

Chapter 10 – Section 2 The relationship is
The hypothesis test calculation and the confidence interval calculation are very similar Not rejecting the hypothesis μ0 is inside the Confidence interval Rejecting the hypothesis μ0 is outside the Confidence interval

Chapter 10 – Section 2 An example of a two-tailed test
A bolt manufacturer claims that the diameter of the bolts average 10.0 mm H0: Diameter = 10.0 H1: Diameter ≠ 10.0 We take a sample of size 40 (Somehow) We know that the standard deviation of this measurement is 0.3 mm The sample mean is mm We’ll use a level of significance α = 0.05 An example of a two-tailed test An example of a two-tailed test A bolt manufacturer claims that the diameter of the bolts average 10.0 mm H0: Diameter = 10.0 H1: Diameter ≠ 10.0

Chapter 10 – Section 2 Do we reject the null hypothesis?
10.12 is 0.12 higher than 10.0 The standard error is (0.3 / √ 40) = 0.047 The confidence interval is ± 1.96 • 0.047, or to 10.21 10.0 is outside (10.03, 10.21) Our conclusion We reject the null hypothesis We have sufficient evidence that the population mean diameter is not 10.0 Do we reject the null hypothesis? 10.12 is 0.12 higher than 10.0 The standard error is (0.3 / √ 40) = 0.047 The confidence interval is ± 1.96 • 0.047, or to 10.21 10.0 is outside (10.03, 10.21)

Chapter 10 – Section 2 An example of a left-tailed test
A car manufacturer claims that the mpg of a certain model car is at least 29.0 H0: MPG = 29.0 H1: MPG < 29.0 We take a sample of size 40 (Somehow) We know that the standard deviation of the population is 0.5 The sample mean mpg is 28.89 We’ll use a level of significance α = 0.05 An example of a left-tailed test An example of a left-tailed test A car manufacturer claims that the mpg of a certain model car is at least 29.0 H0: MPG = 29.0 H1: MPG < 29.0

Chapter 10 – Section 2 Do we reject the null hypothesis?
28.89 is 0.11 lower than 29.0 The standard error is (0.5 / √ 40) = 0.079 The confidence interval limit is • 0.079, or 29.02 29.0 is inside (–∞, 29.02) Our conclusion We do not reject the null hypothesis We have insufficient evidence that the population mean mpg is less than 29.0 Do we reject the null hypothesis? 28.89 is 0.11 lower than 29.0 The standard error is (0.5 / √ 40) = 0.079 The confidence interval limit is • 0.079, or 29.02 29.0 is inside (–∞, 29.02)

Chapter 10 – Section 2 An example of a right-tailed test
A bolt manufacturer claims that the defective rate of their product is at most 1.70 per 1,000 H0: Defect Rate = 1.70 H1: Defect Rate > 1.70 We take a sample of size 40 (Somehow) We know that the standard deviation of the population is .06 The sample defect rate is 1.78 We’ll use a level of significance α = 0.05 An example of a right-tailed test An example of a right-tailed test A bolt manufacturer claims that the defective rate of their product is at most 1.70 per 1,000 H0: Defect Rate = 1.70 H1: Defect Rate > 1.70

Chapter 10 – Section 2 Do we reject the null hypothesis?
1.78 is 0.08 higher than 1.70 The standard error is (0.06 / √ 40) = 0.009 The confidence interval limit is 1.78 – • 0.009, or 1.76 1.70 is outside (1.76, ∞) Our conclusion We reject the null hypothesis We have sufficient evidence that the population mean rate is more than 1.70 Do we reject the null hypothesis? 1.78 is 0.08 higher than 1.70 The standard error is (0.06 / √ 40) = 0.009 The confidence interval limit is 1.78 – • 0.009, or 1.76 1.70 is outside (1.76, ∞)

Chapter 10 – Section 2 Learning objectives
Understand the logic of hypothesis testing Test hypotheses about a population mean with σ known using the classical approach Test hypotheses about a population mean with σ known using P-values Test hypotheses about a population mean with σ known using confidence intervals Understand the difference between statistical significance and practical significance 1 2 3 5 4

Chapter 10 – Section 2 A significant statistical difference is one where the hypothesis test, for equality, is rejected A statistical significance does not necessarily mean that it is practically significant If we have a large sample size, we will be able to pinpoint the rejectable values of the population mean Our analysis may be unnecessarily precise

Summary: Chapter 10 – Section 2
A hypothesis test of means compares whether the true mean is either Equal to, or not equal to, μ0 Equal to, or less than, μ0 Equal to, or more than, μ0 There are three equivalent methods of performing the hypothesis test The classical approach The P-value approach The confidence interval approach

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