Hypothesis Tests for a Population Mean in Practice Chapter 10 Section 3 Hypothesis Tests for a Population Mean in Practice

Chapter 10 – Section 3 Learning objectives Test hypotheses about a population mean with σ unknown 1

Chapter 10 – Section 3 In the previous section, we assumed that the population standard deviation, σ, was known This is not a realistic assumption There is a parallel between Chapters 9 and 10 Sections 9.1 and 10.2 … solving the problems assuming that σ was known Sections 9.2 and 10.3 … solving the problem assuming that σ was not known σ not being known is a much more practical assumption In the previous section, we assumed that the population standard deviation, σ, was known This is not a realistic assumption In the previous section, we assumed that the population standard deviation, σ, was known This is not a realistic assumption There is a parallel between Chapters 9 and 10 Sections 9.1 and 10.2 … solving the problems assuming that σ was known Sections 9.2 and 10.3 … solving the problem assuming that σ was not known

Chapter 10 – Section 3 The parallel between Confidence Intervals and Hypothesis Tests carries over here too For Confidence Intervals We estimate the population standard deviation σ by the sample standard deviation s We use the Student’s t-distribution with n-1 degrees of freedom For Hypothesis Tests, we do the same Use σ for s Use the Student’s t for the normal The parallel between Confidence Intervals and Hypothesis Tests carries over here too The parallel between Confidence Intervals and Hypothesis Tests carries over here too For Confidence Intervals We estimate the population standard deviation σ by the sample standard deviation s We use the Student’s t-distribution with n-1 degrees of freedom

Chapter 10 – Section 3 Thus instead of the test statistic knowing σ we calculate a test statistic using s This is the appropriate test statistic to use when σ is unknown

Chapter 10 – Section 4 We can perform our hypotheses for tests of a population proportion in the same way as when the sample standard deviation is known Two-tailed Left-tailed Right-tailed H0: μ = μ0 H1: μ ≠ μ0 H1: μ < μ0 H1: μ > μ0

Chapter 10 – Section 3 The process for a hypothesis test of a mean, when σ is unknown is Set up the problem with a null and alternative hypotheses Collect the data and compute the sample mean Compute the test statistic The process for a hypothesis test of a mean, when σ is unknown is Set up the problem with a null and alternative hypotheses Collect the data and compute the sample mean The process for a hypothesis test of a mean, when σ is unknown is Set up the problem with a null and alternative hypotheses The process for a hypothesis test of a mean, when σ is unknown is

Chapter 10 – Section 3 Either the Classical and the P-value approach can be applied to determine the significance Classical approach P-value approach

Chapter 10 – Section 3 There are thus only differences between this process and the one using the normal distribution, covered in Section 10.2 We use the sample standard deviation s instead of the population standard deviation σ We use the Student’s t-distribution, with n-1 degrees of freedom, instead of the normal distribution

Chapter 10 – Section 3 An example (the same one as in Section 10.2) A gasoline manufacturer wants to make sure that the octane in their gasoline is at least 87.0 The testing organization takes a sample of size 40 The sample standard deviation is 0.5 The sample mean octane is 86.94 Our null and alternative hypotheses H0: Mean octane = 87 HA: Mean octane < 87 α = 0.05 An example (the same one as in Section 10.2) An example (the same one as in Section 10.2) A gasoline manufacturer wants to make sure that the octane in their gasoline is at least 87.0 The testing organization takes a sample of size 40 The sample standard deviation is 0.5 The sample mean octane is 86.94

Chapter 10 – Section 3 Do we reject the null hypothesis? 86.94 is 0.06 lower than 87.0 The standard error is (0.5 / √ 40) = 0.08 0.06 is 0.75 standard error less The critical t value, with 39 degrees of freedom, is 1.685 –1.685 < –0.75, it is not unusual Our conclusion We do not reject the null hypothesis We have insufficient evidence that the true population mean (mean octane) is less than 87.0 Do we reject the null hypothesis? 86.94 is 0.06 lower than 87.0 The standard error is (0.5 / √ 40) = 0.08 0.06 is 0.75 standard error less The critical t value, with 39 degrees of freedom, is 1.685 –1.685 < –0.75, it is not unusual

Chapter 10 – Section 3 Comparing using the classical approach

Chapter 10 – Section 3 Comparing using the P-value approach

Summary: Chapter 10 – Section 3 A hypothesis test of means, with σ unknown, has the same general structure as a hypothesis test of means with σ known Any one of our three methods can be used, with the following two changes to all the calculations Use the sample standard deviation s in place of the population standard deviation σ Use the Student’s t-distribution in place of the normal distribution