PSY201: Introduction to Quantitative Research in Psychology 1 Lecture Notes PDF

PSY201: Introduction to Quantitative Research in Psychology 1 Lectures: Mondays 9-11 Tutorials: Tuesdays and Wednesdays Instructor: Prof. Keisuke Fukuda (Office hour: Mondays 11-12 @ CCT4067)  Probability Distribution: Discrete Variable Probability Distribution of variable x: a model that describes...

PSY201: Introduction to Quantitative Research in Psychology 1 Lectures: Mondays 9-11 Tutorials: Tuesdays and Wednesdays Instructor: Prof. Keisuke Fukuda (Office hour: Mondays 11-12 @ CCT4067)  Probability Distribution: Discrete Variable Probability Distribution of variable x: a model that describes the long-run behavior of a variable x.  Probability Distribution: Discrete Variable Probability Distribution of variable x: a model that describes the long-run behavior of a variable x. Example: Does your iPod really play favorite? (American Statistician : 263-268) Let’s assume you have 3000 songs in your iPod, of which 50 are made by your favorite artist. When you make a random playlist of 20 songs by shuffling the songs, how likely would the playlist contain the song by your favorite artist? sum of all possibilities = 1 P(x): probability that the 20-song playlist contains x songs by your favorite artist How likely does your random playlist contain at least one song by your favorite artist?  Probability Distribution: Discrete Variable Probability Distribution of variable x: a model that describes the long-run behavior of a variable x. Example: Does your iPod really play favorite? (American Statistician : 263-268) Let’s assume you have 3000 songs in your iPod, of which 50 are made by your favorite artist. When you make a random playlist of 20 songs by shuffling the songs, how likely would the playlist contain the song by your favorite artist? P(x≥1) = 1 -.7138 =.2862 How likely does your random playlist contain at least one song by your favorite artist?  Probability Distribution: Discrete Variable Probability Distribution of variable x: a model that describes the long-run behavior of a variable x.  Probability Distribution: Discrete Variable Probability Distribution of variable x: a model that describes the long-run behavior of a variable x. Example: In a hypothetical country, Any individual is allowed to take a driving test up to 4 times, after which they are not allowed to take the test. Here, let x = the number of attempts made by a randomly selected individual p(x) = a probability that an individual took the test x times. What is the expected value of the number of tests taken by a random individual?  Probability Distribution: Discrete Variable Probability Distribution of variable x: a model that describes the long-run behavior of a variable x. Example: In a hypothetical country, Any individual is allowed to take a driving test up to 4 times, after which they are not allowed to take the test. Here, let  Probability Distribution: Discrete Variable Probability Distribution of variable x: a model that describes the long-run behavior of a variable x.  Probability Distribution: Discrete Variable Probability Distribution of variable x: a model that describes the long-run behavior of a variable x. Example: In a hypothetical TV factory, There are two flat screen suppliers with known distribution of the number of flaws in their product where: x (or y) = number of flaws p(x) (or p(y)) = probability that a flat screen has x (or y) flaws Which supplier would you chose?  Probability Distribution: Discrete Variable Probability Distribution of variable x: a model that describes the long-run behavior of a variable x. Example: In a hypothetical TV factory, Variance is different! Mean is the same!  Probability Distribution: Continuous Variable Probability Distribution of variable x: a model that describes the long-run behavior of a variable x.  Probability Distribution: Continuous Variable Probability Distribution of variable x: a model that describes the long-run behavior of a variable x.  Probability Distribution: Continuous Variable Probability Distribution of variable x: a model that describes the long-run behavior of a variable x.  Probability Distribution: Continuous Variable Probability Distribution of variable x: a model that describes the long-run behavior of a variable x.  Probability Distribution: Continuous Variable Probability Distribution of variable x: a model that describes the long-run behavior of a variable x.  Probability Distribution: Continuous Variable Probability Distribution of variable x: a model that describes the long-run behavior of a variable x.  Probability Distribution: Continuous Variable Probability Distribution of variable x: a model that describes the long-run behavior of a variable x.  Normal distribution Normal distribution: Bell-shaped and symmetric distribution that approximates distributions of many variables in the real world.  Normal distribution Normal distribution: Bell-shaped and symmetric distribution that approximates distributions of many variables in the real world.  Standard Normal Distribution Standard normal distribution (or z curve): A normal distribution with the mean = 0 and the standard deviation = 1 σ=1 σ=1 -1 0 1 (μ)  Standard Normal Distribution Standard normal distribution (or z curve): A normal distribution with the mean = 0 and the standard deviation = 1  Standard Normal Distribution Standard normal distribution (or z curve): A normal distribution with the mean = 0 and the standard deviation = 1 Example: What’s the area more extreme than z = 1.42?  Standard Normal Distribution Standard normal distribution (or z curve): A normal distribution with the mean = 0 and the standard deviation = 1  Standard Normal Distribution Standard normal distribution (or z curve): A normal distribution with the mean = 0 and the standard deviation = 1  Standard Normal Distribution Standard normal distribution (or z curve): A normal distribution with the mean = 0 and the standard deviation = 1 Example: What’s the probability that z exceeds 1.96?  Standard Normal Distribution Standard normal distribution (or z curve): A normal distribution with the mean = 0 and the standard deviation = 1 Example: How do we find the extreme 5% of the distribution? -z* = -1.96 z* = 1.96! Applying standard to other normal distributions More generally,  Applying standard to other normal distributions Example: IQ distribution  Applying standard to other normal distributions Example: How extreme is IQ = 130?  Applying standard to other normal distributions Example: What’s the proportion of population with IQ between 75 and 125? Applying standard to other normal distributions Checking normality of the distribution (a.k.a normal QQ plot) Observed score The set of normal scores depends on the sample size of the observed scores. Normal score Checking normality of the distribution (a.k.a normal QQ plot) Sampling Distribution Example: In an imaginary upper-year seminar course with 20 students, Sampling Distribution Example: In an imaginary upper-year seminar course with 20 students, Let’s consider taking 4 sets of random samples with 5 observations each Sampling Distribution Example: In an imaginary upper-year seminar course with 20 students, If we were to create 50 random samples of n = 5, If we were to do this for all possible combinations of samples, n = 5, we will get the sampling distribution!  Sampling Distribution of sample mean (x) Example: In an imaginary upper-year seminar course with 20 students, sample size = 5 sample size = 10 sample size = 20 μ = 8.25 μ = 8.25 μ = 8.25 As sample size increases, the variability of the sample mean shrinks, and the sample means better approximate the population mean!  Sampling Distribution of sample mean (x) Example: Time to first goal in NHL game (American Statistician : 151-154) sample size = 5 sample size = 10 sample size = 30 μ = 13 μ = 13 μ = 8.25 13 The same holds true even when the population distribution is not normal! Also, as the sample size increases, the sampling distribution better approximates normal distribution!  Sampling Distribution of sample mean (x) General properties of the sampling distributions of x 1. 2.  Sampling Distribution of sample mean (x) General properties of the sampling distributions of x 3. 4.  Sampling Distribution of sample mean (x) General properties of the sampling distributions of x  Sampling Distribution of sample mean (x) When n is large (n ≥ 30) or the population distribution is normal, then the standardized variable (z) = has (at least approximately) a standard normal (z) distribution!  Sampling Distribution of sample mean (x) Example: Fat content of Hot dogs A hotdog company claims that the average fat content of a hotdog is 18 grams with a standard deviation (σ) of 1gram. How do we know the company is telling the truth? 1. Purchase a bag of hotdog (n = 36 hotdogs). 2. Compute the sample mean (x = 18.4 grams) 3. Compute the standard deviation of sample means (σx)  Sampling Distribution of sample mean (x) Example: Fat content of Hot dogs A hotdog company claims that the average fat content of a hotdog is 18 grams with a standard deviation (σ) of 1gram. How do we know the company is telling the truth? 1. Purchase a bag of hotdog (n = 36 hotdogs). 2. Compute the sample mean (x = 18.4 grams) 3. Compute the standard deviation of sample means (σx) 4. Compute how likely we would observe the sample mean if the company’s claim is correct! It’s < 1% chance to have observed the sample mean!  ^ Sampling Distribution of sample proportion (p) sample proportion of successes ( ) Example: In an imaginary college (n = 19000) with 44% female students, the distribution of the sample proportion of female students in 500 samples are: sample size = 10 sample size = 25 sample size = 50 sample size = 100 μ = 0.44 μ = 0.44 μ = 0.44 μ = 0.44  ^ Sampling Distribution of sample proportion (p) sample proportion of successes ( ) Example: Considering an imaginary disease with prevalence of 7%, the distribution of the sample proportion of infected patients in 500 samples are: sample size = 10 sample size = 25 sample size = 50 sample size = 100 sample size = 50 sample size = 100 p = 0.07 p = 0.07 p = 0.07 p = 0.07  ^ Sampling Distribution of sample proportion (p) ^ General properties of the sampling distributions of p 1. 2.  ^ Sampling Distribution of sample proportion (p) ^ General properties of the sampling distributions of p 1. 2. 3.  ^ Sampling Distribution of sample proportion (p) ^ General properties of the sampling distributions of p and  ^ Sampling Distribution of sample proportion (p) Example: For the imaginary disease with prevalence of 7%, a group of researchers claimed that they came up with a potential cure that reduces its prevalence. How do we know if they are telling the truth? 1. Randomly sample 200 individuals who received the cure (n = 200). ^ 2. Compute the sample proportion (p = 6/200 = 0.03) 3. Compute the standard deviation of sample proportion (σ^ p)  ^ Sampling Distribution of sample proportion (p) Example: For the imaginary disease with prevalence of 7%, a group of researchers claimed that they came up with a potential cure that reduces its prevalence. How do we know if they are telling the truth? 1. Randomly sample 200 individuals who received the cure (n = 200). 2. ^ Compute the sample proportion (p = 6/200 = 0.03) 3. Compute the standard deviation of sample proportion (σ^ p) 4. Confirm that the distribution of sample proportion is approximately normal.  ^ Sampling Distribution of sample proportion (p) Example: For the imaginary disease with prevalence of 7%, a group of researchers claimed that they came up with a potential cure that reduces its prevalence. How do we know if they are telling the truth? 1. Randomly sample 200 individuals who received the cure (n = 200). 2. ^ Compute the sample proportion (p = 6/200 = 0.03) 3. Compute the standard deviation of sample proportion (σ^ p) 4. Confirm that the distribution of sample proportion is approximately normal. 5. Compute how likely we would have observed the sample proportion if the treatment had no effect. It’s about 1% chance to have observed the sample proportion!