Tutorial 5 Solutions

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Royal Melbourne Institute of Technology **We aren't endorsed by this school
Course
STATISTICS MISC
Subject
Statistics
Date
Sep 1, 2023
Pages
9
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Week 5 TUTORIAL 5 SOLUTIONS Question 5.1 The lifetimes of the heating element in a Heatfast electric oven are normally distributed, with a mean of 7.8 years and a standard deviation of 2.0 years. The heating element in ovens is guaranteed for two years. a) (i) What percentage of ovens sold will need replacement in the guarantee period because of element failure? Include a statement describing your answer. Let X = lifetime of heating element (years) °~±²7.8, 2 ³ ´ µ²° ¶ 2´ · NORM.DIST(2,7. 8,2,TRUE) · 0.0019 · 0.19% OR µ ¸ ¹º» ¼ ³º½.¾ ³ ¿ · µ²À ¶ Á2.90´ · 0.0019 · 0.19% ²from statistical tables´ 0.19% of the ovens sold will need replacement in the guarantee period. (ii) In a year in which 10,000 ovens are sold, how many ovens would you expect to have to replace in the guarantee period because of element failure? µ²° ¶ 2´ ∗ 10000 · 0.0019 ∗ 10000 · +, -./01 (iii) What proportion of elements are expected to last for between 2 and 10 years? Write a statement explaining your answer.
Week 5 µ²2 ¶ ° ¶ 10´ · 2345. 6789²+:, ;. <, =, 94>?´ Á 2345. 6789²=, ;. <, =, 94>?´ · 0.8625 OR µ D 2 Á 7.8 2 ° Á E F 10 Á 7.8 2 G · µ²Á2.90 ¶ À ¶ 1.10´ = µ²À ¶ 1.10´ Á µ²À ¶ Á2.90´ · 0.8643 Á 0.0019 · 0.8624 ²from statistical tables´ 0.8624 is the proportion of elements that are expected to last between 2 to 10 years. (iv) Heatfast is reconsidering the length of the guarantee period on heating elements. Calculate the length of guarantee period such that Heatfast would expect to replace a maximum of 1% of ovens due to element failure. Include a statement describing your answer. µ²° ¶ J ´ = 0.01 =NORM.INV(0.01,7.8,2) = 3.147 K 3 LMNOP Finding this unknown X value will give the lifetime value for which 1% of ovens are less than (i.e. a maximum of 1%). As the statistical tables are for the Z distribution: µ²À ¶ Q ´ · 0.01 The critical value of Z which cuts off a 1% tail (from the bottom line of Table 2) is 2.327. So, to cut-off 1% in the left-hand tail, the critical value will be z* = -2.327 . You can also use Table 1a. To find the unknown value of X, the following formula is used: ° · E R FÀ Here, J · 7. 8 R ²2. 0 S Á2. 327´ · 3.146 Since a guarantee period is always a round number, the appropriate choice is 3 years.
Week 5 The length of guarantee period such that Heatfast would expect to replace a maximum of 1% of ovens due to element failure is 3 years. (v) How long do 95% of the ovens last? Include a statement describing your answer. µ²° ¶ J ´ · 0.95 TU µ²° > J ´ · 0.05 =NORM.INV(0.95,7.8,2) · 11.09 K 11.1 LMNOP As the statistical tables are for the Z distribution: µ²À ¶ Q ´ · 0.95 TU µ²À > Q ´ · 0.05 Critical value of Z cutting off a 5% right tail (from the bottom line of Table 2) is 1.645. You can also use table 1b. Q ∗· 1.645. To find the unknown value of X, the following formula is used: ° · E R FÀ Here, J ∗· 7. 8 R ²2. 0 S 1. 645´ · 11.09 95% of ovens last at most 11.1 years.
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