That is find 3 x p μ note 2 x p 2 x p 3 x p μ μ μ

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That is, find: ) 3 X ( P - μ < Note: ) 2 X ( P ) 2 X ( P ) 3 X ( P + μ > = - μ < < - μ < This is illustrated in the graph below. PDF of X μ- 3 μ- 2 μ μ +2 Therefore, the answer must be smaller than the probability calculated for part (a).
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Econ 325 – Chapter 6 9 The solution method follows similar steps to part (a): ( ) ) 75 . 0 ( F 1 ) 75 . 0 Z ( P 1 Z P ) X ( se ) 3 ( ) X ( se X P ) 3 X ( P - = < - = - < = μ - - μ < μ - = - μ < 4 3 A look-up in the Standard Normal Distribution Table gives: 0.7734 = ) 75 . 0 ( F The answer is: 0.2266 0.7734 = - = - μ < 1 ) 3 X ( P Econ 325 – Chapter 6 10 (c) What is the probability that the sample mean differs from the population mean by more than 4 hours ? That is, find: lower tail upper tail ) 4 X ( P 2 )] 4 X ( P ) 4 X ( P ) 4 X 4 ( P 1 + μ > = + μ > + - μ < = + μ < < - μ - Follow the steps in part (a) to get: ( ) ) 1 ( F 1 ) 1 Z ( P 1 Z P ) 4 X ( P - = < - = > = + μ > 4 4 A look-up in the Standard Normal Distribution Table gives: 0.8413 = ) 1 ( F The answer is: 0.3174 0.8413) (1 2 = - = + μ > ) 4 X ( P 2
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Econ 325 – Chapter 6 11 (d) Suppose that a second (independent) random sample of 10 students was taken. Without doing the calculations, state whether the probability in part (a) will be higher, lower, or the same for the second sample. The standard error of X is: n ) X ( se σ = An increase in the sample size from n =4 to n =10 gives a smaller standard error. This leads to more concentration about the population mean μ and so ) X ( P 2 + μ > becomes lower . This is illustrated in the graph. PDF of X n = 10 n = 4 μ μ +2 Econ 325 – Chapter 6 12 The Central Limit Theorem The normal distribution is a convenient approximation in many applications. The Central Limit Theorem gives a justification for this. Let the random sample 1 X , 2 X , . . . , n X be a set of random variables that are independently and identically distributed with mean μ and variance 2 σ . The random variables need not follow the normal distribution – they may fit a skewed distribution or any other non-normal distribution. Consider the sample mean: = = n 1 i i X n 1 X Earlier lecture notes stated the standard error of X as: n ) X ( se σ = Define the standardized random variable: n X ) X ( se X Z σ μ - = μ - = The Central Limit Theorem states that as n becomes ‘large’ the distribution of Z approaches the standard normal distribution.
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