Statistics 200 Objectives: Lecture #20 Thursday, October 27, 2016 Textbook: Sections 9.6, 11.1, 11.2 Objectives: • Apply sampling distribution for one sample mean to confidence intervals and hypothesis tests. • Identify situations in which t-multipliers and t-tests should be used instead of z-multipliers and z-tests.

We have begun a strong focus on Inference Means Proportions One population mean One population proportion Two population proportions Difference between Means Mean difference This week

Categorical data (2 categories) Quantitative data parameter: statistic: parameter: statistic:

Clicker Question: Consider the following three survey questions: Do you plan to vote in the upcoming presidential election? How old are you? Which candidate do you most support? How many of these questions will produce Quantitative data? 1 2 3

Example 1: The population is normally distributed.

Clicker Question Which statement(s) are false, when comparing the original distribution to the two sampling distributions All three distributions have the same value for the mean As the sample size increases, the standard deviation for the sampling distribution decreases The original distribution will always have a smaller standard deviation than what is found with either of the two sampling distributions The sampling distributions suggest possible values for the population mean.

Example 1: The population is normally distributed.

Example 1: The population is normally distributed.

Sampling Distribution: Standard Deviation & Standard Error Statistic Standard Deviation We generally do not know p. Thus, we don’t know s.d.(p-hat). Similarly: We generally do not know σ. Thus, we don’t know s.d.(x-bar).

Sampling Distribution: Standard Deviation & Standard Error Statistic Standard Deviation Standard Error: Estimated St Dev if p is unknown, use: Here, s is the sample standard deviation. if σ is unknown, use:

Example 1: The population is normally distributed. Substitute s for σ:

Standard normal (dotted red) vs. t (solid black) Degrees of freedom for t distribution: 1, 5, and 20 (as d.f. increases, the t looks more like the standard normal.)

Standard normal (dotted red) vs. t (solid black) Degrees of freedom for t distribution: 1, 5, and 20 We worry a lot about teaching t vs. z, but the difference is tiny for degrees of freedom usually seen in practice.

Confidence Interval Formula sample estimate ± (margin of error) sample estimate ± (multiplier × standard error) Generic Formula: Specific for Population Mean: µ This template slide is for a slide from the middle of a class presentation. To follow this template, you should craft a sentence headline that states an assertion you want to make about your topic. In this design, having no assertion translates to having no slide. In the body of the slide, you should support the sentence-headline assertion visually with photographs, drawings, diagrams, equations, or words arranged visually. Use supporting text only where necessary. Do not rely on bulleted lists, because bulleted lists do not reveal the connections between details. See Shaw et al. [1996] This slide shows one orientation for the image and supporting text. Other orientations exist, as depicted in the sample slides that follow. This design of slides has been shown [Alley, Schreiber, and Muffo, 2005] to be more effective than the traditional topic-subtopic design at having students understand and recall key information from class lectures. References: Alley, Michael, Madeline Schreiber, and John Muffo, "Pilot Testing of a New Design of Presentation Slides to Teach Science and Engineering," 2005 Frontiers in Education Conference, paper 1213 (Indianapolis, IN: ASEE/IEEE, October 2005). Shaw, Gordon, Robert Brown, and Philip Bromiley, “Strategic Stories: How 3M Is Rewriting Business Planning,” Harvard Business Review (May–June, 1998), pp. 41–50 Here, t* depends on confidence level and df = (n – 1).

Multipliers: from the t table (not a complete list, obviously) Conf. level: 0.90 0.95 0.98 0.99 1 df 6.31 12.71 31.82 63.66 2 df 2.92 4.30 6.96 9.92 3 df 2.35 3.18 4.54 5.84 9 df 1.83 2.26 2.82 3.25 20 df 1.72 2.09 2.53 2.85 30 df 1.70 2.04 2.46 2.75 Infinite df 1.645 1.96 2.326 2.576

Clicker Question: What kind of variable is this? Categorical Quantitative Example 2: We ask each of 31 students “how many regular ‘text’ friends do you have?” Survey results: n = 31 X-bar = 6 friends s = 2.0 friends Calculate a 95% Confidence Interval: How can we estimate the population mean number of regular “text” friends for all STAT 200 students using these data?

Confidence Interval Formula sample estimate ± (margin of error) sample estimate ± (multiplier × standard error) Generic Formula: Survey results: n = 31 X-bar = 6 friends s = 2.0 friends This template slide is for a slide from the middle of a class presentation. To follow this template, you should craft a sentence headline that states an assertion you want to make about your topic. In this design, having no assertion translates to having no slide. In the body of the slide, you should support the sentence-headline assertion visually with photographs, drawings, diagrams, equations, or words arranged visually. Use supporting text only where necessary. Do not rely on bulleted lists, because bulleted lists do not reveal the connections between details. See Shaw et al. [1996] This slide shows one orientation for the image and supporting text. Other orientations exist, as depicted in the sample slides that follow. This design of slides has been shown [Alley, Schreiber, and Muffo, 2005] to be more effective than the traditional topic-subtopic design at having students understand and recall key information from class lectures. References: Alley, Michael, Madeline Schreiber, and John Muffo, "Pilot Testing of a New Design of Presentation Slides to Teach Science and Engineering," 2005 Frontiers in Education Conference, paper 1213 (Indianapolis, IN: ASEE/IEEE, October 2005). Shaw, Gordon, Robert Brown, and Philip Bromiley, “Strategic Stories: How 3M Is Rewriting Business Planning,” Harvard Business Review (May–June, 1998), pp. 41–50 Thus, the 95% CI is

Confidence Interval Interpretation We are 95% confident that the… sample mean sample proportion population mean population proportion range of values for the …number of regular “text” friends for STAT 200 students is between 5.3 and 6.7 friends. Calculated Interval: 6.0 ± 0.7 friends (5.3 to 6.7 friends) This template slide is for a slide from the middle of a class presentation. To follow this template, you should craft a sentence headline that states an assertion you want to make about your topic. In this design, having no assertion translates to having no slide. In the body of the slide, you should support the sentence-headline assertion visually with photographs, drawings, diagrams, equations, or words arranged visually. Use supporting text only where necessary. Do not rely on bulleted lists, because bulleted lists do not reveal the connections between details. See Shaw et al. [1996] This slide shows one orientation for the image and supporting text. Other orientations exist, as depicted in the sample slides that follow. This design of slides has been shown [Alley, Schreiber, and Muffo, 2005] to be more effective than the traditional topic-subtopic design at having students understand and recall key information from class lectures. References: Alley, Michael, Madeline Schreiber, and John Muffo, "Pilot Testing of a New Design of Presentation Slides to Teach Science and Engineering," 2005 Frontiers in Education Conference, paper 1213 (Indianapolis, IN: ASEE/IEEE, October 2005). Shaw, Gordon, Robert Brown, and Philip Bromiley, “Strategic Stories: How 3M Is Rewriting Business Planning,” Harvard Business Review (May–June, 1998), pp. 41–50

Confidence Interval Conclusion 95% C.I.: 5.3 to 6.7 friends In the population, we may conclude, with 95% confidence, that on average, STAT 200 students have A. more than 6 friends. more than 4 friends. fewer than 5 friends. fewer than 6 friends. This template slide is for a slide from the middle of a class presentation. To follow this template, you should craft a sentence headline that states an assertion you want to make about your topic. In this design, having no assertion translates to having no slide. In the body of the slide, you should support the sentence-headline assertion visually with photographs, drawings, diagrams, equations, or words arranged visually. Use supporting text only where necessary. Do not rely on bulleted lists, because bulleted lists do not reveal the connections between details. See Shaw et al. [1996] This slide shows one orientation for the image and supporting text. Other orientations exist, as depicted in the sample slides that follow. This design of slides has been shown [Alley, Schreiber, and Muffo, 2005] to be more effective than the traditional topic-subtopic design at having students understand and recall key information from class lectures. References: Alley, Michael, Madeline Schreiber, and John Muffo, "Pilot Testing of a New Design of Presentation Slides to Teach Science and Engineering," 2005 Frontiers in Education Conference, paper 1213 (Indianapolis, IN: ASEE/IEEE, October 2005). Shaw, Gordon, Robert Brown, and Philip Bromiley, “Strategic Stories: How 3M Is Rewriting Business Planning,” Harvard Business Review (May–June, 1998), pp. 41–50

Example 3: The population is NOT normally distributed.

Are all sampling distributions normal? _____ When do we have to be cautious? with _____ sample sizes where the original population is not ______ in shape No small normal One-Sample t procedure is valid if one of the conditions for normality is met: Sample data suggest a normal shape We have a large sample size (n ≥ __) or 30 Sampling distribution will look normal in shape

If you understand today’s lecture… 9.61, 9.62, 9.64, 9.65, 11.25, 11.30, 11.32, 11.33 Objectives: • Apply sampling distribution for one sample mean to confidence intervals and hypothesis tests. • Identify situations in which t-multipliers and t-tests should be used instead of z-multipliers and z-tests.