statistics normal distribution

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  1. Use the standard normal proportions table to calculate the following probabilities based on the standard normal curve. In each case, sketch a normal curve and shade in the appropriate area.

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2. What formula do you use when calculating the number of standard deviations from the mean of a random variable x for a normal distribution problem?

3. The National Collegiate Athletic Association (NCAA) requires Division I athletes to score at least 820 on combined mathematics and verbal parts of the SAT exam to compete in their first college year. (Higher scores are required for students with poor high school grades.) In 2002, the scores of the 1.3 million students taking the SATs were approximately Normal with mean 1020 and standard deviation 207.

a. Sketch a normal curve and label the mean and specific x values.

b. To determine what percent of all students that had a score less than 820, what formula do you need?

c. Substitute relevant values into the formula and solve for z.

d. Use the probability notation involving the inequality in relation to z to find the probability.

4. Red Blood Cell Count Let x = red blood cell count in millions per cubic millimeter of whole blood. For healthy females, x has an approximately normal distribution with mean pastedGraphic_5.png and standard deviation pastedGraphic_6.png.

a.) Sketch a normal curve and label the mean and specific x values.

b.) Determine the probability of a woman having a red blood cell count less than 4.35 million per cubic millimeter.

c.) What is the probability of a woman having a red blood cell count greater than 5.56 million per cubic millimeter?

5. According to data from the National Health Survey the heights of adult women have a mean of pastedGraphic_7.png inches and a standard deviation of pastedGraphic_8.pnginches.

a. Sketch a normal curve and label the mean and specific x values.

b. The U.S. Army requires women’s heights to be between 58 and 80 inches. Find the percentage of women meeting that height requirement. Are many women being denied the opportunity to join the Army because they are too short or too tall? (Hint: For z-values too large for the chart use an approximation of 1 for its probability).

c. In order to fit into a Russian Soyuz spacecraft, an astronaut must have a height between 64.5 and 72 inches. What percentage of women meet that requirement?

5.) Budget Maintenance The amount of money spent weekly on cleaning, maintenance, and repairs at a large restaurant was observed over a long period of time to be approximately normally distributed, with a mean pastedGraphic_9.png and standard deviationpastedGraphic_10.png.

a. If $646 is budgeted for next week, what is the probability that the actual costs will exceed the budgeted amount?

b. Inverse Normal Distribution How much should be budgeted for weekly repairs, cleaning, and maintenance so that the probability that the budgeted amount will be exceeded in a given week is only 0.10?

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