# Critical Value and Probabilities

4. (1) Assume that the readings on the thermometers are normally distributed with a mean of 0o and standard deviation of 1.00o C. A thermometer is randomly selected and tested. Draw a diagram and find the probability in degrees.

Between -1.18 and 2.15

(2) The lengths of pregnancies are normally distributed with a mean of 268 days and a standard deviation of 15 days. A woman claims to give birth 308 after conception.

a. Find probability of a pregnancy lasting 308 days or longer. What does the result suggest?

b. If we stipulate that a premature birth is in the lowest 4%, find the length that separates premature from full term births.

(3) Ages 62, 46, 68, 64, 57. Assume the samples of size 2 are randomly selected with replacement from the population of five ages.

a. after identifying the 25 different possible samples, find the mean of each of them.

b. Describe sampling distribution of means.

c. Find the mean of the sampling distribution

d. Is the mean of the sampling distribution (from c.) equal to the mean of the five ages above? Are those always equal?

(4) Use the Central Limit Theorem - Assume normal distribution with a mean of 172 lb and standard deviation of 29 lb

a. 4 men are randomly selected, what is the probability that they have a mean weight between 160 and 180 lb.

b. Why can the central limit theorem be used in (a.), even though the sample size does not exceed 30?

(5) Building a bench to seat 18 men, normal distribution with a mean of 14.4 in and std deviation of 1.0 in hip breadth

a. What is the minimum length of the bench if you want a 0.975 probability that it will fit the combined hip breadths of 18 randomly selected men?

b. What would be wrong with actually using the result form part (a.) as the bench length?

(6) Estimate the probability of getting at least 65 girls in 100 births. Assume boys and girls are equally likely. Is it unusual to get at least 65 girls in 100 births?

(7) Construct Normal Quantile Plots, identify if the corresponding z scores that are used for a normal quantile plot the construct the normal Qplot and determine whether the data appear to be from a population with normal distribution.

a. LA Lakers heights, use the sample in inches 85, 79, 82, 73, 78

5. (1) Find critical value zo/2 that corresponds with the given confidence level of 92%

(2) Find the margin of error , assume the sample is used to estimate a population proportion. Find the margin for E.

a. n=1200, x=400, 99% confidence

(3) Of 491 randomly selected adults 65% favor the death penalty

a. find the point estimate of the % of adults who are in favor of the death penalty.

b. find a 95% confidence interval estimate of the % of adults who are in favor.

c. can we safely conclude that the majority of adults are in favor? Explain why.

(4) Women heights are normally distributed with a mean of 63.6 in. and a std deviation of 2.5 in.

a. how many women must be surveyed if we want to estimate the % who are taller than 5ft? assume that we want 98% confidence that the error is no more that 2.5 % points. (answer is substantially smaller than 2172)

(5) Margin of error and confidence Int.,

a. ages of drivers in the passing lane while driving 25 mi/h with left signal flashing: 99% confidence; n=50, mean=80.5 yrs and std is 4.6 yrs

(6) Find confidence Int.,

a. 99% confidence; n=15, mean=496, s=0.07in

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