class: center, middle, inverse, title-slide # Confidence Intervals and
the t Distribution ## Cohen Chapter 6
.small[EDUC/PSY 6600] --- class: center, middle ## “It is common sense to take a method and try it. <br> If it fails, admit it frankly and try another. <br> But above all, try something.” ### -- Franklin D. Roosevelt --- <!-- Research By Design: Why We Might Not Have Statistics Without Guinness Brewery – A History of the t-Test (2.5 min)--> <iframe width="1000" height="750" src="https://www.youtube.com/embed/U9Wr7VEPGXA?controls=0&start=2" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe> --- # Problems with z-tests Usually we do **NOT** know `\(\sigma\)`, so we can **NOT** compute .nicegreen[*Standard Error for the Mean (SEM)*] `\((SE_{\bar{x}})\)` -- .large[ $$ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} $$ ] -- Can you use the .dcoral[sample's *SD*] `\((s)\)` in place of .dcoral[populations's *SD*] `\((\sigma)\)` to calculate the .nicegreen[SEM] `\((SE_{\bar{x}})\)` as part of the `\(z\)`-test? -- - .nicegreen[Large samples] – Yes (N > 300 participants) - .nicegreen[Small samples] – No, **inaccurate** results -- .large[ $$ z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}} \xrightarrow{\quad \text{If N is large} \quad} z= \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}} $$ ] --- # Sample Size Matters .pull-left[ ## .dcoral[Small] samples As samples get .coral[smaller]: `\(N \downarrow\)` - the skewness of the sampling distribution of `\(s \uparrow\)` - `\(s\)` .nicegreen[**under**estimates] `\(\sigma\)` - `\(z\)` will `\(\uparrow\)` - an .nicegreen[**over**estimate] `\(\uparrow\)` risk of **Type I error** ] -- .pull-right[ ## .dcoral[LARGE] samples Compared to smaller samples: - `\(s\)` **un**biased estimate of `\(\sigma\)` - `\(\sigma\)` is a constant, *unknown truth* - `\(s\)` is NOT a constant, since it varies from sample to sample - As `\(N\)` increases, `\(s \rightarrow \sigma\)` ] --- background-image: url(figures/fig_will_gossett.png) # The t Distribution, “student’s t” .pull-right[ 1908, William Gosset - Guinness Brewing Company, England - Invented `t-test` for **small** samples for brewing quality control Wrote paper using moniker .nicegreen[“a student”] discussing nature of `\(SE_M\)` when .dcoral[using] `\(s^2\)` .dcoral[instead of] `\(\sigma^2\)` - Worked with Fisher, Neyman, Pearson, and Galton [Priceonomics: The Guinness Brewer Who Revolutionized Statistics](https://priceonomics.com/the-guinness-brewer-who-revolutionized-statistics/) ] --- # Student’s t Distributions <img src="figures/fig_t_vs_z.png" width="75%" style="display: block; margin: auto;" /> --- # Student’s t & Normal Distributions .pull-left[ .large[.nicegreen[**Similarities between t & z**]] - Follows mathematical function - Symmetrical, continuous, bell-shaped - Continues to `\(\pm\)` infinity - Mean: `\(M = 0\)` - Area under curve = `\(p(event[s])\)` - When `\(N\)` is **large** `\((\approx 300)\)`, `\(t = z\)` ] -- .pull-right[ .large[.dcoral[**How t is Different from z**]] - Family of distributions - Different distribution for each `\(N\)` (or `\(df\)`) - Larger area in **tails** (%) for any value of `\(t\)` corresponding to `\(z\)` - `\(t_{cv} \gt z_{cv}\)`, for a given `\(\alpha\)` - More difficult to reject `\(H_0\)` w/ t-distribution - `\(df = N - 1\)` - As `\(df \uparrow\)`, the critical value of `\(t \rightarrow z\)` ] --- class: inverse ## Interactive Visulatization [Understanding the t-distribution and its normal approximation](https://rpsychologist.com/d3/tdist/) <img src="figures/app_t_viz.png" width="50%" style="display: block; margin: auto;" /> > Dr. Kristoffer Magnusson, aka "R Psychologist" > Centre for Psychiatry Research, Karolinska Institutet, Stockholm, Sweden --- <img src="figures/fig_t_table_top.png" width="80%" style="display: block; margin: auto;" /> --- <!-- StatCast: What is a t-test? (10 min)--> <iframe width="1000" height="750" src="https://www.youtube.com/embed/0Pd3dc1GcHc?controls=0&start=2" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe> --- # Calculating the t-Statistic .bluer[ `\(x\)` is interval/ratio data (ordinal okay: `\(\ge 10-16\)` levels or values) ] Like `\(z\)`, the `\(t\)`-statistic represents a **SD** score (the # of SE's that `\(\bar{x}\)` deviates from `\(\mu\)`) .large[ .center[$$t = \frac{\bar{x} - \mu_x}{\frac{s_x}{\sqrt{N}}}$$] .center[$$df = N - 1$$] ] When `\(\sigma\)` is known, `\(t\)`-statistic is sometimes computed .bluer[(rather than `\(z\)`-statistic)] if `\(N\)` is **small** .center[.large[.dcoral[ Estimate the population `\(SEM\)` with sample data: ]]] .center[ Estimated `\(SEM\)` is the amount a sample's observed **mean** <br>may have deviated from <br> the true or population value <br>just due to random chance variation due to sampling. ] --- # Assumptions of a 1-sample t-test .large[**1. Sample was drawn at .dcoral[RANDOM]** *(at least as representative as possible)*] - Nothing can be done to fix a NON-representative samples! - Can .bluer[**NOT**] statistically test <br> -- .large[**2. .dcoral[SD] of the sampled population = .dcoral[SD] of the comparison population**] - Nearly impossible to check, can .bluer[**NOT**] statistically test <br> -- .large[**3. Variable has a .dcoral[NORMAL] distribution in the population**] - .bluer[**NOT**] as important if the sample is large, due to the **Central Limit Theorem** -- - .bluer[**CAN**] statistically test: - Visual inspection of a .nicegreen[histogram], .nicegreen[boxplot], and/or .nicegreen[QQ plot] *(straight 45 degree line)* - Calculate the Skewness & Kurtosis... less clear guidelines - Conduct .nicegreen[Shapiro-Wilks] test *(p < .05 ??? not normal)* --- ## Formula Sheet .dcoral[**One-Sample Tests**] <img src="figures/fig_formulas_1sample_mean.png" width="100%" style="display: block; margin: auto;" /> --- ## Ex) 1-Sample t Test: mean vs. *historic control* A physician states that, .dcoral[in the past], the average number of times he saw each of his patients during the year .dcoral[**5**]. However, he believes that his patients have visited him significantly **more frequently** during the past year. In order to validate this statement, he .nicegreen[randomly selects **10** of his patients] and determines the number of office visits during the past year. He obtains the values presented to the below. .center[.nicegreen[**9, 10, 8, 4, 8, 3, 0, 10, 15, 9**]] Do the data support his contention that the average number of times he has seen a patient in the last year is .dcoral[**different** that 5]? -- .pull-left[ **Step 1) State the Hypotheses ** $$ H_0: \mu = 5 $$ $$ H_1: \mu \ne 5 $$ **Step 2) Select the Statistical Test and Significance Level** * 1-sample t-Test for the Mean * TWO tailed & `\(\alpha = .05\)` ] -- .pull-right[ <img src="figures/fig_ttest_ex1a.jpg" width="80%" style="display: block; margin: auto;" /> ] --- ## Ex) 1-Sample t Test: mean vs. *historic control* **Step 3) Select the Sample and Collect the Data** ```r x <- c(9, 10, 8, 4, 8, 3, 0, 10, 15, 9) # compare to historic value: mu = 5 ``` -- .pull-left[ .dcoral[**Sample Size (*N*)**] ```r length(x) ``` ``` [1] 10 ``` .dcoral[**Sample Mean (*M*)**] $$ \bar{X} = \frac{\sum{X}}{N} $$ ```r mean(x) ``` ``` [1] 7.6 ``` ] -- .pull-right[ .dcoral[**Sample Standard Deviation (*SD*)**] ```r sd(x) ``` ``` [1] 4.247875 ``` .dcoral[**Standard Error (*SEM*)**] $$ \sigma_{\bar{X}} = \frac{s}{\sqrt{N}} $$ ```r SE <- sd(x) / sqrt(length(x)) SE ``` ``` [1] 1.343296 ``` ] --- ## Ex) 1-Sample t Test: mean vs. *historic control* **Step 4) Find the Region of Rejection** <img src="figures/fig_t_table_top_df9.jpg" width="62%" style="display: block; margin: auto;" /> --- ## Ex) 1-Sample t Test: mean vs. *historic control* **Step 4) Find the Region of Rejection** <img src="figures/fig_t_table_top_df9_2tail_05.jpg" width="62%" style="display: block; margin: auto;" /> --- ## Ex) 1-Sample t Test: mean vs. *historic control* **Step 4) Find the Region of Rejection** <img src="figures/fig_t_table_top_df9_2tail_05_cv.jpg" width="62%" style="display: block; margin: auto;" /> --- ## Ex) 1-Sample t Test: mean vs. *historic control* .pull-left[ **Step 5) Calculate the Test Statistic** ] .pull-right[ <img src="figures/fig_ttest_ex1a.jpg" width="100%" style="display: block; margin: auto;" /> ] --- ## Ex) 1-Sample t Test: mean vs. *historic control* .pull-left[ **Step 5) Calculate the Test Statistic** ] .pull-right[ <img src="figures/fig_ttest_ex1b.jpg" width="100%" style="display: block; margin: auto;" /> ] --- ## Ex) 1-Sample t Test: mean vs. *historic control* .pull-left[ **Step 5) Calculate the Test Statistic** .dcoral[**Observed t-score (*t*)**] ```r t <- (mean(x) - 5)/ SE t ``` ``` [1] 1.935538 ``` ] .pull-right[ <img src="figures/fig_ttest_ex1c.jpg" width="100%" style="display: block; margin: auto;" /> ] --- ## Ex) 1-Sample t Test: mean vs. *historic control* .pull-left[ **Step 5) Calculate the Test Statistic** .dcoral[**Observed t-score (*t*)**] ```r t <- (mean(x) - 5)/ SE t ``` ``` [1] 1.935538 ``` ] .pull-right[ <img src="figures/fig_ttest_ex1d.jpg" width="100%" style="display: block; margin: auto;" /> ] --- ## Ex) 1-Sample t Test: mean vs. *historic control* **Calculate the p-value** *Only can get a rough range with this table* <img src="figures/fig_t_table_top_df9_find_194.jpg" width="62%" style="display: block; margin: auto;" /> --- ## Ex) 1-Sample t Test: mean vs. *historic control* **Calculate the p-value** *Only can get a rough range with this table* <img src="figures/fig_t_table_top_df9_found_194.jpg" width="62%" style="display: block; margin: auto;" /> --- ## Ex) 1-Sample t Test: mean vs. *historic control* A physician states that, .dcoral[in the past], the average number of times he saw each of his patients during the year .dcoral[**5**]. However, he believes that his patients have visited him significantly **more frequently** during the past year. In order to validate this statement, he .nicegreen[randomly selects **10** of his patients] and determines the number of office visits during the past year. He obtains the values presented to the below. .center[.nicegreen[**9, 10, 8, 4, 8, 3, 0, 10, 15, 9**]] Do the data support his contention that the average number of times he has seen a patient in the last year is .dcoral[different that 5]? **Step 6) State the Conclusion** *APA format in context* -- > Even though this sample of .coral[ten] patients has a mean of .bluer[**7.60**] visits per year, this could be due to sampling variance and does not provide evidence patients have .bluer[**changed**] the mean number of visit per year.nicegreen[, *t*(9) = 1.94, .05 < *p* < .10]. -- *Note: Using the t **table**, we can only get a rough idea of the p-value, but when we use **software** we will get a p-vlaue with lots of decimal places.* --- <!-- Dr. Nic: Understanding Confidence Intervals: Statistics Help (4 min) --> <iframe width="1000" height="750" src="https://www.youtube.com/embed/tFWsuO9f74o?controls=0&start=2" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe> --- # Confidence Intervals .large[Statistics are .dcoral[point estimates] of a *population parameter* with .dcoral[**with error**]] How .nicegreen[**close**] is estimate to population parameter? - Confidence interval (CI) around point estimate .nicegreen[*(Range of values)*] - Upper limit: UL or UCL - Lower limit: LL or LCL Confidence Intervals express our .nicegreen[**confidence**] or .nicegreen[**uncertainty**] in a .dcoral[sample statistic's] ability to estimate the .dcoral[population parameter] based on the .nicegreen[*width*], which depends on both the sample's spread `\((SEM)\)` and critical balue `\((z_{CV} \text{ or } t_{cv})\)` - Both are function of `\(N\)` - Larger `\(N \rightarrow\)` Narrower CI <br><br> - More confident that sample point estimate (statistic) approximates population parameter - .nicegreen[Narrow CI:] Less confidence, more precision *(less error)* - .nicegreen[Wide CI:] More confidence, less precision *(more error)* --- <!-- Joshua Emmanuel: Confidence Intervals - Introduction (3.5 min) --> <iframe width="1000" height="750" src="https://www.youtube.com/embed/MbXThbTSrVI?controls=0&start=2" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe> --- <!-- Dr. Nic: Calculating the Confidence interval for a mean using a formula - statistics help (5.5 min) --> <iframe width="1000" height="750" src="https://www.youtube.com/embed/s4SRdaTycaw?controls=0&start=2" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe> --- <!-- Crash Course Statistics #20: Confidence Intervals (13 min) --> <iframe width="1000" height="750" src="https://www.youtube.com/embed/yDEvXB6ApWc?controls=0&start=2" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe> --- class: inverse ## Interactive Shiny App .large[Confidence Intervals for a Sample Mean, A Simulation] [link](https://shiny.rit.albany.edu/stat/confidence/) <img src="figures/app_ci_simulate.png" width="60%" style="display: block; margin: auto;" /> > University at Albany Psychology Department, Psychology Department, by Bruce Dudek --- class: inverse ## Interactive HTML App .large[Statistical Applet: Confidence Intervals] [link](http://digitalfirst.bfwpub.com/stats_applet/stats_applet_4_ci.html) <img src="figures/app_ci_applet.png" width="50%" style="display: block; margin: auto;" /> > BFW-Bedford, Freeman & Worth Publishing, by Digital First --- class: inverse ## Interactive Visulatization [Interpreting Confidence Intervals](https://rpsychologist.com/d3/ci/) <img src="figures/app_ci_viz.png" width="60%" style="display: block; margin: auto;" /> > Dr. Kristoffer Magnusson, aka "R Psychologist" > Centre for Psychiatry Research, Karolinska Institutet, Stockholm, Sweden --- # The Formula for all Confidence Intervals Every .nicegreen[Confidence Interval] has two parts: * Point Estimate Value = our best guess * Margin of Error = plus-or-minus uncertainty -- $$ \text{Point Estimate Value} \quad \pm \quad \text{Margin of Error} $$ -- Every .nicegreen[Margin of Error] has two parts: * Critical Value = look up on a table for a certain % confidence * Standard Error for the Estimate = measure of spread $$ \text{Point Estimate Value} \quad \pm \quad \text{Critical Value} \quad \times \quad \text{Standard Error for the Estimate} $$ -- .pull-left[ .coral[Known population *SD* &/or Large *N*] `$$\bar{x} \; \pm \; z_{CV} \times \frac{\sigma}{\sqrt{N}}$$` ] -- .pull-right[ .coral[Unknown population *SD* &/or Small *N*] `$$\bar{x} \; \pm \; t_{CV} \times \frac{s}{\sqrt{N}}$$` ] --- # Steps to Construct a Confidence Interval .pull-left[ 1. Select your random sample size <br><br> 2. Select the .coral[**Level of Confidence**] <br><br> - Generally 95% *(can be 80, 90, or even 99%)* <br><br> 3. Select random sample and collect data <br><br> 4. Find the .coral[**Critical Value**] <br><br> - Based on `\(\alpha = 1 - Conf\)` & # of tails - Default: 95% `\((\alpha = .05)\)` and 2 tails<br><br> 5. Calculate the Interval **End Points** `$$Est \pm CV_{Est} \times SE_{Est}$$` ] -- .pull-right[ .pull-left[.nicegreen[ Narrow CI: - large smaple - Lower % ]] .pull-right[ .nicegreen[ Wider CI: - smaller sample - Higher % ]] <br><br> .large[.dcoral[95% CI with z score]] `$$\bar{x} \pm 1.96 \times \frac{\sigma}{\sqrt{N}}$$` .large[.dcoral[99% CI with z score]] `$$\bar{x} \pm 2.58 \times \frac{\sigma}{\sqrt{N}}$$` ] --- ## Formula Sheet .dcoral[**One-Sample Tests**] <img src="figures/fig_formulas_1sample_mean.png" width="100%" style="display: block; margin: auto;" /> --- ## Ex) Confidence Interval for a Mean A physician states that, .dcoral[in the past], the average number of times he saw each of his patients during the year .dcoral[**5**]. However, he believes that his patients have visited him significantly **more frequently** during the past year. In order to validate this statement, he .nicegreen[randomly selects **10** of his patients] and determines the number of office visits during the past year. He obtains the values presented to the below. .center[.nicegreen[**9, 10, 8, 4, 8, 3, 0, 10, 15, 9**]] Construct a .dcoral[95% confidence interval] for the mean number of visits per patient. --- ## Ex) Confidence Interval for a Mean **Find the Critical Value:** for 95%, use `\(\alpha\)` = .05, but ALWAYS use TWO_TAILED for confidence intervals! <img src="figures/fig_t_table_top_df9_2tail_05_cv.jpg" width="62%" style="display: block; margin: auto;" /> --- ## Ex) Confidence Interval for a Mean ```r x <- c(9, 10, 8, 4, 8, 3, 0, 10, 15, 9) ``` .pull-left[ .dcoral[**Sample Size (*N*)**] ```r length(x) ``` ``` [1] 10 ``` .dcoral[**Sample Mean (*M*)**] ```r mean(x) ``` ``` [1] 7.6 ``` .dcoral[**Sample Standard Deviation (*SD*)**] ```r sd(x) ``` ``` [1] 4.247875 ``` ] -- .pull-right[ Standard Error: .dcoral[**Standard Error (*SEM*)**] ```r SE <- sd(x) / sqrt(length(x)) SE ``` ``` [1] 1.343296 ``` .dcoral[**Confidence Interval (t_CV = 2.262)**] ```r mean(x) - 2.262 * SE ``` ``` [1] 4.561464 ``` ```r mean(x) + 2.262 * SE ``` ``` [1] 10.63854 ``` ] --- background-image: url(figures/fig_ex_t_ci.png) ## Ex) Confidence Interval for a Mean A physician states that, in the past, the average number of times he saw each of his patients during the year was `\(5\)`. However, he believes that his patients have visited him significantly **more frequently** during the past year. In order to validate this statement, he randomly selects `\(10\)` of his patients and determines the number of office visits during the past year. He obtains the values presented to the below. .center[.nicegreen[**9, 10, 8, 4, 8, 3, 0, 10, 15, 9**]] Construct a .dcoral[95% confidence interval] for the mean number of visits per patient. --- # Estimating the Population Mean .pull-left[ .nicegreen[Point estimate (M) is in the center of CI] Degree of confidence determined by `\(\alpha\)` and corresponding critical value (CV) - Commonly use 95% CI, so `\(\alpha = .05\)` - Can also compute a .90, .99, or any size CI .dcoral[z-distribution:] <br>Known population variance or N is large (about 300) `$$\bar{x} \pm z_{cv} \times \frac{\sigma}{\sqrt{N}}$$` .dcoral[t -distribution:] <br>Do not know population variance or N is small `$$\bar{x} \pm t_{cv} \times \frac{s}{\sqrt{N}}$$` ] -- .pull-right[ .large[**is NOT the meaning of a 95% CI**]<br> There is **NOT** a 95% chance that the population M lies between the 2 CLs from your sample’s CI !!! Each random sample will have a different CI with different CLs and a different M value <br> .large[** IS the meaning of a 95% CI**]<br> 95% of the CIs that could be constructed over repeated sampling will contain Μ Yours **MAY** be one of them 5% chance our sample’s 95% CI does not contain `\(\mu\)` Related to **Type I Error** ] --- # APA Style Writeup .large[.nicegreen[**Z-test**]] <br> *(happens to be a statistically significant difference)* <br> The hourly fee .coral[(M = $72)] for our sample of current psychotherapists is significantly greater.nicegreen[, z = 4.0, p < .001,] than the 1960 hourly rate .coral[(M = $63, in current dollars)]. <br> -- .large[.nicegreen[**T-test**]] <br> *(happens to not quite reach .05 significance level)* <br> Although the mean hourly fee for our sample of current psychotherapists was considerably higher .coral[(*M* =$72, *SD* = $22.5)] than the 1960 population mean .coral[(M = $63, in current dollars)], this difference only approached statistical significance.nicegreen[, *t*(24) = 2.00, *p* = .061]. --- class: inverse, center, middle # Let's Apply This to the Cancer Dataset --- # Read in the Data ```r library(tidyverse) # Loads several very helpful 'tidy' packages library(haven) # Read in SPSS datasets library(furniture) # Nice tables (by our own Tyson Barrett) library(psych) # Lots of nice tid-bits ``` ```r cancer_raw <- haven::read_spss("cancer.sav") ``` -- ### And Clean It ```r cancer_clean <- cancer_raw %>% dplyr::rename_all(tolower) %>% dplyr::mutate(id = factor(id)) %>% dplyr::mutate(trt = factor(trt, labels = c("Placebo", "Aloe Juice"))) %>% dplyr::mutate(stage = factor(stage)) ``` --- ## The Cancer Dataset
--- # 1 sample t Test vs. Historic Control > Do the patients .dcoral[weigh] more than .dcoral[165] pounds at intake, on average? ```r cancer_clean %>% dplyr::pull(`weighin`) %>% t.test(mu = 165) ``` -- ``` One Sample t-test data: . t = 2.0765, df = 24, p-value = 0.04872 alternative hypothesis: true mean is not equal to 165 95 percent confidence interval: 165.0807 191.4793 sample estimates: mean of x 178.28 ``` -- > The patients in this study .coral[(*N* = 25)] weigh .bluer[**178.28**] pounds on average, which is significantly more that .bluer[**165**] pounds.nicegreen[, *t*(24) = 2.08, *p* = .049, 95% *CI* [165.08, 191.48]]. --- ## ...Change the Confidence Level > Find a .dcoral[99%] confidence level for the population mean weight. ```r cancer_clean %>% dplyr::pull(`weighin`) %>% t.test(mu = `165`, conf.level = `0.99`) ``` -- ``` One Sample t-test data: . t = 2.0765, df = 24, p-value = 0.04872 alternative hypothesis: true mean is not equal to 165 99 percent confidence interval: 160.3927 196.1673 sample estimates: mean of x 178.28 ``` -- > The patients in this study .dcoral[(*N* = 25)] weigh .bluer[**178.28**] pounds on average, which is significantly more that .bluer[**165**] pounds.nicegreen[, *t*(24) = 2.08, *p* = .049, 99% *CI* [160.39, 196.17]]. --- ## ...Restrict to a Subsample > Do the patients with .dcoral[stage 3 & 4] cancer weigh more than .dcoral[165] pounds at intake, on average? ```r cancer_clean %>% dplyr::filter(`stage %in% c("3", "4")`) %>% dplyr::pull(`weighin`) %>% t.test(mu = `165`) ``` -- ``` One Sample t-test data: . t = 0.82627, df = 5, p-value = 0.4463 alternative hypothesis: true mean is not equal to 165 95 percent confidence interval: 137.0283 219.4717 sample estimates: mean of x 178.25 ``` -- > The patients in this study with stage 3 or 4 cancer .dcoral[(*n* = 6)] weigh .bluer[**178.25**] pounds on average, which is not significantly more that .bluer[**165**] pounds.nicegreen[, *t*(5) = 0.83, *p* = .446, 95% *CI* [137.02, 219.47]]. --- class: inverse, center, middle # Questions? --- class: inverse, center, middle # Next Topic ### Independent Samples t Tests for Means