Lab_Wk06_2023 (1)

STAT251 Fundamentals of Biostatistics LABORATORY NOTES, Week 6 Normal Distribution, t-distribution and Confidence Intervals Aim: The aim of this lab is to determine probabilities and quantiles for the normal distribution, determine and interpret confidence intervals for population means. The normal and t-distributions are required for some of these intervals. Note: Starred (*) exercises do not require Jamovi. 1. The Normal Distribution Mean μ and standard deviation σ. 1.1 Finding probabilities using the properties of the normal curve. Log book questions* : 1. For the following questions:  match the interval with the appropriate diagram of the shaded area under the Normal curve; and  use the diagram above to calculate the area shaded. i. Mean = 0 and sd = 1; between -1 and 3. ii. mean = 0 and sd = 1; between -1 and 1. iii. mean = 55 and sd = 4; between 47 and 59. a) iv. μ = 30 and σ = 5; between 30 and 35. v. μ = 100 and σ = 15; less than or equal to 100. vi. μ = 546.6 and σ = 73.1; between 619.7 and 692.8. b) c) d) e) f)

1.2 The Standard Normal Distribution The Standard Normal distribution has a mean μ = 0 and a standard deviation σ =1. To calculate probabilities from the normal distribution in Jamovi, we need the distrACTION module. Instructions: 1. Click on distrACTION > Normal distribution . 2. Specify the mean and standard deviation under Parameters. 3. To compute the probability P(X ≤ x 1 ), tick the Compute probability option. 4. Specify the value x 1 . 5. The resulting probability will appear on the right hand side output under Results . Alternatively, you can also use the online calculator in Appendix A to calculate probabilities from the normal distribution. Log book questions : 2. For the following questions:  Illustrate on the curve the area represented by the probability. Note: If using Word, use the Insert -> Shapes function, draw the line to outline the region of interest, and use a shape (e.g., a star) to mark the area you need. Alternatively, print these pages out and draw and shade by hand.  Find the probability using Jamovi, an online calculator, or the Standard Normal Distribution Tables (on the Moodle site under Probability Calculators and Statistical Tables ). Reference: derived from those in the Stat131 Laboratory Manual compiled by Assoc. Prof. Anne Porter. 2

1.3 Standardising values and finding probabilities for a given distribution Two steps are involved: 1. Standardise the x values to find the corresponding z-score: z = x − μ σ 2. Use the z -score to find the probabilities. Log book questions : 3. The right hand span of males is normally distributed with a mean of 27 cm and standard devia- tion of 4.5 cm. Let X denote the right hand span of males. For parts (a) to (e), complete the fol- lowing steps:  Determine the z -score.  Illustrate on the curve the area represented by the expression.  Determine the probability requested. a. P (X < 22.5) = Z b. P ( X > 22.5) = Z c. P ( X < 19.5) = Z 4

d. P (19.5 < X < 22.5) = Z e. P ( X < 19) or P ( X > 23) = Z 1.4 Finding raw scores for a given percentile Calculating raw scores for a given percentile is basically the “reverse” of what we did in Section 1.2 and 1.3. Given some random variable X , we are interested in finding out, for example, what is the value x such that 70% of X values lie below x ? In other words, what is the value x such that P( X ≤ x ) = 0.7? Instructions: a. In Jamovi, go to the distrACTION module and select the distribution you are interested in (normal or t-distribution). b. Select Compute quantile(s) , and choose Cumulative quantile . c. Specify the probability P( X ≤ x ) (the percentile divided by 100). Visually, this is the area under the normal curve from its left tail up to the value x that we want to find. d. The value of x will appear on the right hand side output under Results. Log book questions : 4. Given that X , the random variable blood pressure for women aged 29 to 40 years, has a mean of 130 and a standard deviation of 10, answer the following questions: 5

2. Interpreting statistical tables – finding z and t values Instructions for finding probabilities from the t-distribution: 1. Click on distrACTION > T-distribution . 2. Specify the degrees of freedom (df) under Parameters. Ignore the lambda (λ) parameter. 3. To compute the probability P(X ≤ x 1 ), tick the “Compute probability” option under Function. 4. Specify the value x 1 . 5. The result will appear on the right hand side output under Results. Log book questions*: 5. Use Jamovi (or the Standard Normal and t-distribution tables, if you prefer) to find the values marked by question marks in the following diagrams. Remember that the t-distribution with degrees of freedom df = ∞ corresponds to the Standard Normal Distribution. a. t, df = 9 b. Standard Normal Optional: Check your calculations for (a) and (b) in using an online calculator (Appendix A). 3. Confidence intervals for the mean A confidence interval is an interval containing the most plausible values for a parameter . The probability that the interval contains the parameter is the confidence level, e.g. 0.95. The following formulae for Confidence Intervals for a population mean were taken from the lecture notes for convenience. Population Parameter Population Description Estimator Standard error of the estimator Confidence interval µ (σ known) Normally distributed ´ x σ √ n ´ x± z 1 − α / 2 ⋆ σ √ n µ (σ unknown estimated by s ) Normally distributed ´ x s √ n ´ x±t df = n − 1,1 − α / 2 ⋆ s √ n Not Normal and n large ´ x s √ n ´ x±t df = n − 1,1 − α / 2 ⋆ s √ n Not Normal and n small ´ x 7

Open the MathScienceTest data file, available from Moodle. The columns Q1 to Q14 contain the marked responses of individual students to 14 questions on a test (Q1 up to Q14), where 1 means correct and 0 means incorrect. The variable Total is the sum of these marks . In the following section, we will first exam- ine the normality of the Total variable, and then calculate the confidence interval for the mean Total marks. 3.1 Checking Normality It is important to always plot the data first using simple plots such as a histogram (look for a unimodal and bell-shaped plot) and a boxplot (look for symmetry and outliers). Several methods can be used in Jamovi to assess normality. 3.1.1 Graphical methods for checking normality Graphical methods for assessment of possible non-Normality include the histogram and the box and whisker plot. These two plots reveal different aspects of the distribution. In the histogram - look for a unimodal and bell-shaped plot without large gaps. In the boxplot - look for a symmetrical plot with no outlying or extreme values. For an ideal normal distribution, the Normal Quantile-Quantile (Q-Q) Plot will show the points lying exactly on a straight (diagonal) line.  Click on Analysis → Exploration → Descriptives and put Total in the Variables box.  Click on the Plots menu and tick the Histogram , Box plot and Q-Q plot options. 8

Scroll down your sample25 column set to see the randomly chosen values. Logbook questions: 7. For the sample of size n = 25, a. Go to Analysis > Exploration > Descriptives , put sample25 into the Variables box, and record the appropriate values in the table below. You will need the distrACTION module or an online calculator (Appendix A) to determine the multiplier column. Then calculate the confidence intervals in the last column using the formulae at the top of Section 3 . Parameter Estimator ( ´ x va lue) Standard Error ( σ √ n or s √ n value) Degrees of freedom ( n-1 ) 1-α t n − 1,1 − α / 2 ⋆ or z 1 − α / 2 ⋆ multiplier Confidence interval µ (σ unknown) 0.95 µ (σ unknown) 0.99 b. Determine the CI widths and comment upon how interval width varies with confidence level. (NB: CI width is Upper limit - Lower limit). c. To confirm that your calculations are correct, go to Analyses > T-Tests > One-Sample T-test , and put sample25 into the Dependent Variables box. Then under Additional Statistics select Mean Difference and Confidence interval for Mean . Specify the desired level of confidence, and compare the resulting confidence intervals against the ones you calculated. 8. Repeat the Jamovi steps to obtain a sample size of n = 50. Using the output for your sample size of n = 50: a. record the appropriate values in the table below. Parameter Estimator ( ´ x va lue) Standard Error ( σ √ n or Degrees of freedom ( n-1 ) 1-α t n − 1,1 − α / 2 ⋆ or z 1 − α / 2 ⋆ multiplier Confidence interval 10

s √ n value) µ (σ unknown) 0.95 µ (σ unknown) 0.99 b. Determine the CI widths and compare to those obtained in Q7b. What happens to the interval widths as the sample size increases? Explain. Appendix A : Calculating probabilities using a web calculator Probabilities from the normal distribution can also be computed using this applet: https://homepage.divm - s.uiowa.edu/~mbognar/applets/normal.html And probabilities from the t distribution can be computed using https://homepage.divms.uiowa.edu/~mbognar/applets/t.html Other web calculators are available under the Probability Calculators and Statistical Tables section on Moodle. 11