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Normal probabilities

Solve the first problem 17 problems

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1. Empirical rule: values within three deviations ●○○○○ 10 ptsThe values of a bell-shaped (normal) distribution have mean $89$ and standard deviation $14$. Using the empirical rule, find the approximate percentage of values between $47$ and $131$. Round to one decimal place.
2. Normal density at a point ●○○○○ 10 ptsFind $P(x)$ for the normal distribution with mean $\mu = 2$ and standard deviation $\sigma = 3$ at $x = 5$, using $$P(x)=\frac{1}{\sqrt{2\pi\sigma^2}}\,e^{-(x-\mu)^2/(2\sigma^2)}.$$ Use $\pi \approx 3.14$. Round to four decimal places.
3. Diastolic pressure below 70 ●○○○○ 10 ptsDiastolic blood pressures for men aged 35 to 44 are normally distributed with mean 80 mm Hg and standard deviation 10 mm Hg. What is the probability that a randomly chosen man in this age group has a diastolic blood pressure less than 70 mm Hg? Round to two decimal places.
4. Mean of a normal distribution ●●○○○ 20 ptsLet $X$ be normally distributed. Given that $P(-3<X<-1)=P(3<X<5)$, find the expectation $E(X)$.
5. Percentage of pitch times in an interval ●●○○○ 20 ptsTommy Wait's times between pitches are normally distributed with a mean of 36 seconds and a standard deviation of 2.5 seconds. What percentage of his times between pitches are between 30 and 34 seconds? Give the answer as a percentage, rounded to two decimal places.
6. Mean, spread and 90th percentile of heights ●●○○○ 20 ptsFor women athletes at UCLA, heights are approximately normal. The 25th percentile height is $62.5$ inches and the 75th percentile height is $65.5$ inches. Using $z=\pm 0.67$ for these quartiles, find the mean $\mu$ and the standard deviation $\sigma$ of the heights. Then, using $z=1.28$ for the 90th percentile, find the 90th percentile height. Round $\sigma$ and the 90th percentile height to two decimal places.
7. Proportion exceeding a rounded threshold ●●●○○ 50 ptsThe tensile strength of a metal component is normally distributed with mean $10000$ kg/cm² and standard deviation $100$ kg/cm². Each measurement is recorded to the nearest $50$ kg/cm². A measurement counts as exceeding $10150$ when its recorded value is greater than $10150$. What proportion of the measurements exceed $10150$ kg/cm²? Round to four decimal places.
8. Male weight exceeds twice female weight ●●●○○ 50 ptsThe weights $X$ kg of male birds are normally distributed with mean $4.8$ and standard deviation $0.2$. The weights $Y$ kg of female birds are normally distributed with mean $2.7$ and standard deviation $0.15$. $X$ and $Y$ are independent. Find the probability that the weight of a randomly chosen male bird is more than twice the weight of a randomly chosen female bird. Give your answer correct to four decimal places.
9. Martian species height difference probability ●●●○○ 50 ptsSpecies 1 heights are normally distributed with mean 32 and variance 60. Species 2 heights are normally distributed with mean 22 and variance 70. A sample of 10 members of Species 1 and a sample of 14 members of Species 2 are drawn independently. Find the probability that the sample mean of Species 1 exceeds the sample mean of Species 2 by 5 or more. Round to three decimal places.

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