5.5. Suppose that X has mean m and variance ² (a) Prove Cantelli's inequality PIX-mzals avo.
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![5.5. Suppose that X has mean m and variance o
(a) Prove Cantelli's inequality
P[X-mza]s
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- A newspaper article reported that people spends a mean of 6 hours per day watching tv, with a standard deviation of 1.8 hours. A psychologist would like to conduct interviews with the 15% of the population who spend the most time watching tv. She assumes that the daily time spent watching tv is normally distributed. At least how many hours of daily tv watching are necessary to be eligible for the interview? Carry intermediate computations to 4 places, and the answer to 1.2. Suppose that at a gas station the daily demand for regular gasoline is normally distributed with a mean of 1000 gallons and a standard deviation of 100 gallons. The station manager has just opened the station for business and there is exactly 2300 gallons of regular gasoline in storage. The next delivery is scheduled later tomorrow at the close of business. (a) Let X1 and X2 denote the demand of today and tomorrow, respectively. Let X = X1 + X2. Find the distribution of X. (b) Find the probability that the manager will have enough regular gasoline to satisfy today’s and tomorrow’s demand.Show that if ν > 2, the chi-square distribution has arelative maximum at x = ν − 2. What happens whenν = 2 or 0 <ν< 2?
- Suppose that f (x) = (3/2)x2 for −1< x < 1. Determine the mean and standard deviation of X.Suppose that a group of researchers is planning to test a new weight loss supplement. They have selected a random sample of 45 people who are trying to lose weight and plan to measure the amount of weight lost after one month of using the supplement. Assume that the researchers know from prior experiments that the standard deviation of weight lost in one month, , is 1.7 lb. To show that the supplement is effective, they plan to use a one-sample z‑test of Ho : u= 0lb against H1 : u > 0lb , where is the mean amount of weight lost in one month. They have also determined that, for a test with a significance level of 0.05, the power of the test is 0.9347 if the mean amount of weight lost is actually 0.8 lb. What is the probability that the researchers will reject their null hypothesis if the mean amount of weight lost is 0.8 lb or more? Give your answer as a percentage, precise to two decimal places.Suppose that X has a Poisson distribution with h = 80. Find theprobability P(x is less than or equal to 80)
- Suppose that you enter a fantasy baseball league. Suppose that the 2021 team budget, say , is randomly drawn from a uniform distribution on the interval , where the unit is U.S. million dollars. In addition, suppose that after the value has been observed , the 2022 team budget, say , is randomly drawn from a uniform distribution on the interval . In other words, the 2022 budget is at most as large as the 2021 budget. a) For any given value of x(50<x<350), obtain E[Y|X=x] b) In view of part (a), obtain E[Y|X] c) Atlanta Braves won the 2021 World Series title. Their estimated 2022 payroll is about $130 million. Would your 2022 fantasy baseball budget be on average larger than their 2022 payroll? Explain briefly.Suppose we want to know what proportion p of students in a largeuniversity have studied calculus in high school. At least how many studentsshould we include in a sample if we want to estimate p with an error bound of 0.03 (at most) with 95% confidence ?Assume that the readings at freezing on a batch of thermometers are normally distributed with a mean of 0°C and a standard deviation of 1.00°C. A single thermometer is randomly selected and tested.If 4.7% of the thermometers are rejected because they have readings that are too high and another 4.7% are rejected because they have readings that are too low, find the two readings that are cutoff values separating the rejected thermometers from the others.Interval of acceptable thermometer readings _________ °CEnter your answer as an interval (a,b) with 2-decimal place accuracy.
- Suppose that 95% of the bags of certain fertilizer mix weigh between 49 and 53 pounds. Averages of three succesive bags were plotted, and 47.5% of these were observed to lie between 51 and X pounds. Estimate the value of X. State assumptions you make and say whether these assumptions are likely to be true for this example.2. When using the central limit theorem for means with n = 67, it is not necessary to assume the distribution of the population data is normally distributed. True FalseAssume that the readings at freezing on a bundle of thermometers are normally distributed with a mean of 0°C and a standard deviation of 1.00°C. A single thermometer is randomly selected and tested.If 4.8% of the thermometers are rejected because they have readings that are too high and another 4.8% are rejected because they have readings that are too low, find the two readings that are cutoff values separating the rejected thermometers from the others.interval of acceptable thermometer readings =_____________ °C
