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10) Suppose that the actual amount of instant coffee that a filling machine puts into "125 c.c. "cups is a normal random variable with a mean \( \mu \), and a standard deviation \( \sigma=5.102 \) c.c. If \( 2.5 \% \) of the cups overflow, what percentage of the cups will contain less than 110 c.c. a) \( 2.5 \% \) b) \( 16.35 \% \) c) \( 33.65 \% \) d) None of these 11) A firm is placing three different orders for supplies among three different distributors. Each order is randomly assigned to one of the distributors, and a distributor can receive more than one order. Find the probability that all the three orders go to the same distributor. \( \begin{array}{llll}\text { a) } \frac{6}{27} & \text { b) } \frac{1}{27} & \text { c) } \frac{3}{27} & \text { d) None of these }\end{array} \) 12) The members of a consulting firm rent cars from two cars rental agencies, \( 60 \% \) from agency \( X_{\text {and }} 40 \% \) from agency \( Y \). It is known that \( 10 \% \) of the cars rented from agency \( X \) need a tune-up, and \( \% \) the consulting firm was found to need a tune-up, what is the probability that it was rented from agency \( X \) ? \( 0.25 \) brobability that \( \begin{array}{lll}\text { a. } 0.75 & \text { b) } 0.25 & \text { c) } 0.08\end{array} \)
7) \( A \), and \( B \) are two independent events. If \( P(A)=0.6, P(A \cup B)=1 \), then \( P(A \cap f) \) equals a) \( 0.6 \) b) \( 0.0 \) c) \( 0.16 \) d) None of these 8) A pair of fair dice are rolled till getting sum \( =7 \). Let \( X \) a random variable representing the number of rolls required to get sum 7 for the first time. What is \( P(X>2) \) ? a) \( 0.0067 \) b) \( 0.0278 \) c) \( 0.6944 \) d) None of these 9) Suppose that \( X \) is an exponential random variable. If \( P(X \geq 10)=0.1353 \), then \( P(10 \leq \) \( X \leq 15 \) ) equals approximately a) \( 0.1353 \) b) \( 0.0608 \) c) \( 0.0855 \) d) None of these
20) Seprese thur \( 50 \$ 5 \) of the \( \mathrm{NDU} \) students have GPA \( \geq 2.75 \). If \( 100 \mathrm{NDU} \) students were mading selveded approximate the probability that at most 55 of them will have a) 03643 b) \( 0.3528 \) c) \( 0.3413 \) d) None of these A random sample of size \( n=25 \) selected from a normal population with unknown (30 \( u \) and a known variance \( \sigma^{2}=9 \), shows \( \bar{x}=21.25 \), If we set \( \mu=\bar{x}=21.25 \), then We are \( 95 \% \) confident that the maximum error of estimate equals \( 1.176 \) We are \( 90 \% \) confident that the maximum error of estimate equals \( 0.987 \) Both parts (a) and (b) are true None of these is true
18) If \( X \) and \( Y \) be two continuous random variables with joint probability density function \[ f(x, y)=\left\{\begin{array}{lr} \frac{1}{2} & \text { for } 0\frac{P \text {-vatue }}{2} \) b) Reject the claim since the given value of \( a

p \) - value 25) We have a normal population with unknown mean \( \mu \) and unknown variane \( a^{1} \). Suppose we are testing \( H_{0}: \mu=55 \) against \( H_{1}: \mu \neq 55 \quad \) at \( \alpha=0.05 \) If random sample of size 16 shows \( x=57 \) and \( S^{2}-15.5 \), then we should a. Reject \( H_{0} \), since O.V.T.S \( =7.75 \) and is not between \( -2.131 \) and \( 2.131 \) b. Accept \( H_{0} \), since O.V.T.S \( =1.96 \) and is between \( -2.201 \) and \( 2.201 \) d. Reject \( H_{0} \), since O.V.T.S \( =2.032 \) and is not between \( -1.96 \) and \( 1.96 \) 26) Suppose we have a normal population with a known mean \( \mu=55 \), and unknown variance \( \sigma^{2} \). We are testing the claim that \( \sigma=6 \). If a random sample of size 16 selected from the population shows \( \bar{x}=40 \), and \( S=7 \), then the value of O.V.T.S equals a) \( 10.417 \) b) \( 5.714 \) c) \( 20.417 \) d) None of these 21) In testing the claim \( \mu=107 \), type I error happens when we a) Reject the claim \( \mu=107 \), when actually the claim is not true b) Accept the claim \( \mu=107 \), when actually the claim is true c) Reject the claim \( \mu=107 \), when actually the claim is true d) Accept the claim \( \mu-407 \), when actually the claim is not true 22) Suppose we are testing \( H_{0}: \mathrm{P}=0.69 \), against \( H_{1}: \mathrm{P} \neq 0.69 \quad \) at \( \alpha=0.05 \) A random sample of 220 were selected. The sample shows \( \hat{p}=0.75 \). What is the \( P \) - value of the test? a) \( 0.0548 \) b) \( 0.0274 \) c) \( 0.0239 \) d) \( 0.0478 \) 23) Suppose we have a normal population with unknown variance \( \sigma^{2} \), and we are testing \( H_{0}: \sigma^{2}=25 \) against \( H_{a}: \sigma^{2}<25 \quad \) at \( \alpha=0.05 \) If a random sample of size 11 was selected and the observed value of the test statistic "O.V.T.S" was calculated to be \( 3.29 \), then we shall a) Accept \( \mathrm{H}_{0} \), since O.V.T.S \( <18.307 \) b) Reject \( H_{0} \), since O.V.T.S \( <3.94 \) c) Accept \( H_{0} \), since O.V.T.S \( <4.575 \) d) Reject \( H_{0} \), since O.V.T.S \( <18.307 \) Page 12 of 1 b) The samplesieentemath c) The sample viov in is large d) The sample siee it is small and the population standard deviation \( \sigma \) is unknown 28) Sugwow that the GPI of the \( \mathrm{NDU} / \) mudenis is known to have a mean of \( 2.9 \), and a standard deviation of \( 0.12 \). If a random sample of 30 students was selected, then the Truhabition a) 0 (M)o b) \( 0.4504 \) e) \( 0.0 \mathrm{NOH} \) d) None of the above 20) Suppose that \( 50 \% \) of the NDU students have GPA 2 2.75. If \( 100 \mathrm{NDU} \) students were randomly selected, approximate the probability that at most 55 of them will have \( G P A \leq 2.75 \) a) \( 0.3643 \) b) \( 0.3528 \) c) \( 0.3413 \) d) None of these 12) A random sample of size \( n=25 \) selected from a normal population with unknown (30 mean \( \mu \) and a known variance \( \sigma^{2}=9 \), shows \( x=21 \), 25, If we set \( \mu=x=21.25 \), then a) We are \( 95 \% \) confident that the maximum error of estimate equals \( 1.176 \) b) We are \( 90 \% \) confident that the maximum error of estimate equals \( 0.987 \) c) Both parts (a) and (b) are true d) None of these is true