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**Unformatted text preview: **7.15 Given a normal distribution with m=100 and =10, if you select a sample of n=25, what is the probability that x is a. Less than 95? b. Between 95 and 97.5? c. Above 102.2? d. There is a 65% chance that X is above what value? Solution: , = Sample Size n 25 , Sample Mean = = m 100 , = = = Sample Standard Deviation n 1025 2 a. = For x 95 b. c. z- = score x- =- =- . 95 1002 2 5 d. e. &lt; = &lt;- . = . - . = . Px 95 PZ 2 5 0 5 0 4938 0 0062 f. [- ] from z distribution table g. . Probability that x is less than 95 is 0 0062 h. i. = For x 95 j. k. z- = score x- =- =- . 95 1002 2 5 l. = . For x 97 5 m. n. z- = score x- = . - =- . 97 5 1002 1 25 o. &lt; &lt; . = - . &lt; &lt;- . = .- . = . P95 x 97 5 P 2 5 Z 1 25 0 4938 0 3944 0 0994 p. . . Probability that x is lies between 95 and 97 5 is 0 0994 q. = . For x 102 2 r. s. z- = score x- = . - = . 102 2 1002 1 1 t. u. &gt; . = &gt; . = . - . = . Px 102 2 PZ 1 1 0 5 0 3643 0 1357 v. [- ] from z distribution table w. . . Probability that x is above 102 2 is 0 1357 x. y. . + = . 0 5 y 0 65 z. , = .- . = . Or y 0 65 0 5 0 15 aa. ab. = - .- z 0 36 from z distribution table ac. ad.- =- . x 1002 0 39 ae. , = . Or x 99 22 af. ag. Therefore, there is a 65% chance that x is above 99.22. ah. ai. aj. ak. al. am.7.21 an. Time spent using e-mail per session is normally distributed, with m = 8 minutes and =2 minutes. If you select a random sample of 25 sessions, ao. a. what is the probability that the sample mean is between 7.8 and 8.2 minutes? ap. b. what is the probability that the sample mean is between 7.5 and 8 minutes? aq. c. If you select a random sample of 100 sessions, what is the probability that the sample mean is between 7.8 and 8.2 minutes? ar. d. Explain the difference in the results of (a) and (c). as. at. Solution: au. av. aw. , = Sample Size n 25 ax. ay. , Sample Mean = = m 8 az. ba. , = = = . Sample Standard Deviation n 225 0 4 bb. a. = . For x 7 8 b. c. z- = score x- = . - . =- . 7 8 80 4 0 5 d. = . For x 8 2 e. f. z- = score x- = . - . = . 8 2 80 4 0 5 g. . &lt; &lt; . = - . &lt; &lt; . = . + . = . P7 8 x 8 2 P 0 5 Z 0 5 0 1915 0 1915 0 383 h. . . . Probability that x is lies between 7 8 and 8 2 is 0 383 i. j. = . For x 7 5 k. l. z- = score x- = . - . =- . 7 5 80 4 1 25 m. = For x 8 n. o. z- = score x- = - . = 8 80 4 0 p. . &lt; &lt; = - . &lt; &lt; = . + = . P7 5 x 8 P 1 25 Z 0 0 3944 0 0 3944 q. . . Probability that x is lies between 7 5 and 8 is 0 3944 r. s. t. , = Sample Size n 100 u. v. , Sample Mean = = m 8 w. x. , = = = . Sample Standard Deviation n 2100 0 2 y. z. = . For x 7 8 aa. ab. z- = score x- = . - . =- 7 8 80 2 1 ac. = . For x 8 2 ad. ae. z- = score x- = . - . = 8 2 80 2 1 af. . &lt; &lt; . = - &lt; &lt; = . + . = . P7 8 x 8 2 P 1 Z 1 0 3413 0 3413 0 6826 ag. . . . Probability that x is lies between 7 8 and 8 2 is 0 6826 ah....

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