University:
Jackson College
Course:
MAT-010 | Basic Mathematics
Academic year:

2024
Views:

185

Pages:

4
Author:

Bennett Wheeler

Question 1 Which would you expect to have a density curve that is higher at the mean: the standard normal distribution or a normal distribution with standard deviation 0.5? Explain. Answer 1 A normal distribution with a smaller standard deviation (0.5) is more concentrated around the mean, making the density curve higher at the mean compared to the standard normal distribution (which has a standard deviation of 1). The smaller spread means data points are closer to the mean, resulting in a taller peak. Question 2 A normal distribution μ = 30 μ=30 and σ = 5 σ=5. Find the z-score corresponding to x = 25 x=25. Answer 1 Given: μ = 30 , σ = 5 , x = 25 μ=30,σ=5,x=25 Using the z-score formula: z = x − μ σ = 25 − 30 5 = − 1.0 z= σ x−μ = 5 25−30 =−1.0 So, the z-score for x = 25 x=25 is − 1.0 −1.0. Question 3 For X ∼ N ( 335 , 50 ) X∼N(335,50), find the z-score corresponding to an observation of 284. Answer 1 Given: μ = 335 , σ = 50 , x = 284 μ=335,σ=50,x=284 Using the z-score formula: z = x − μ σ = 284 − 335 50 = − 51 50 = − 1.02 z= σ x−μ = 50 284−335 = 50 −51 =−1.02 Thus, the z-score for an observation of 284 is − 1.02 −1.02. Question 4 True or false: a. All normal distributions are symmetrical b. All normal distributions have a mean of 1.0 c. All normal distributions have a standard deviation of 1.0 d. The total area under the curve of all normal distributions is equal to 1 Answer 1 a. True: All normal distributions are symmetrical around the mean. b. False: Normal distributions can have any mean value, not necessarily 1.0. c. False: Normal distributions can have different standard deviations, not always 1.0. d. True: The total area under the curve of a normal distribution is always equal to 1, as it represents the probability. Question 5 Explain why t t-distributions tend to be flatter and more spread out than the normal distribution. Answer 1 The t t-distribution has more variability, especially with fewer degrees of freedom, causing it to be flatter and more spread out than the normal distribution. As degrees of freedom increase, the t t-distribution approaches the normal distribution shape.