Chapter 6: Problem 55
Find the following probabilities for the standard normal random variable:
a. \(P(.3
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Consider a binomial random variable with \(n=45\) and \(p=.05 .\) Fill in the blanks below to find some probabilities using the normal approximation. a. Can we use the normal approximation? Calculate \(n p=\) ______ and \(n q=\) ______ b. Are \(n p\) and \(n q\) both greater than \(5 ?\) Yes_____ No______ c. If the answer to part \(b\) is yes, calculate \(\mu=n p=\) _______ and \(\sigma=\sqrt{n p q}=\) _____ d. To find the probability of 10 or fewer successes, what values of \(x\) should be included? \(x=\) _______ e. To include the entire block of probability for the first value of \(x=\) ______, start at ______. f. Calculate \(z=\frac{x \pm .5-n p}{\sqrt{n p q}}=\) _______. g. Calculate \(P(x \leq 10) \approx P(z<\)_____ ) \(=\) ______.
There is a difference in sports preferences between men and women, according to a recent survey. Among the 10 most popular sports, men include competition- type sports-pool and billiards, basketball, and softball-whereas women include aerobics, running, hiking, and calisthenics. However, the top recreational activity for men was still the relaxing sport of fishing, with \(41 \%\) of those surveyed indicating that they had fished during the year. Suppose 180 randomly selected men are asked whether they had fished in the past year. a. What is the probability that fewer than 50 had fished? b. What is the probability that between 50 and 75 had fished? c. If the 180 men selected for the interview were selected by the marketing department of a sporting goods company based on information obtained from their mailing lists, what would you conclude about the reliability of their survey results?
a. Find a \(z_{0}\) such that \(P\left(z>z_{0}\right)=.9750 .\) b. Find a \(z_{0}\) such that \(P\left(z>z_{0}\right)=.3594\).
a. Find a \(z_{0}\) such that \(P\left(-z_{0}
Students very often ask their professors whether they will be "curving the grades." The traditional interpretation of "curving grades" required that the grades have a normal distribution, and that the grades will be assigned in these proportions: $$ \begin{array}{l|lllll} \text { Letter Grade } & \mathrm{A} & \mathrm{B} & \mathrm{C} & \mathrm{D} & \mathrm{F} \\ \hline \text { Proportion of Students } & 10 \% & 20 \% & 40 \% & 20 \% & 10 \% \end{array} $$ a. If the average "C" grade is centered at the average grade for all students, and if we assume that the grades are normally distributed, how many standard deviations on either side of the mean will constitute the "C" grades? b. How many deviations on either side of the mean will be the cutoff points for the "B" and "D" grades?
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