1. Suppose the commuting time from Georgetown to ...



Chapter 8 Continuous Probability Models

Things You Should Know

1. Types of continuous probability distributions

Positively Skewed Negatively Skewed Bimodal

2. Probabilities and Uniform Distribution

a) Uniform distribution is a straight horizontal line across the graph. The total area under the graph is 1. To calculate the height of the horizontal line [pic]. To find a given probability, calculate the rectangular area of that section.

3. Properties of the Normal Distribution

a) 68% of data values of X lies between the range of [pic] and[pic].

95% of the data values of X lies between the range of [pic] and[pic].

99.7% of the data values of X lies between the range of range [pic] and[pic].

b) Calculate a z-score using the formula: [pic]

c) To calculate the probability of a continuous variable, use the calculated z-score and the z-score table. Remember [pic]and [pic]

• Use the graphing calculator function normalcdf(lower, upper, mean, std. dev.)

d) To find a cut-off value substitute the cut-off for the z-score and work backwards to find X.

• Use the graphing calculator function Norminv(cut-off, mean, std. dev.)

4. Normal Sampling and Modeling

a) If a sample of a population has been given use the sample mean and sample standard deviation, then use the z-score method.

[pic] [pic] [pic]

b) If discrete data is being used it is permissible to use the standard z-score, but remember to use a continuity correction. i.e. [pic]

5. Normal Approximation to the Binomial Distribution

a) A normal approximation can be made with discrete data if [pic]and [pic].

b) Calculate: [pic] and [pic] then use the z-score test

6. Hypothesis Testing

a) Determine the null hypothesis: [pic]

b) State the alternative hypothesis: [pic]

c) Conduct the hypothesis test (calculate the z-score)

d) If the probability is greater than the significance level, accept[pic], otherwise reject[pic].

7. Confidence Intervals

a) Using the equation [pic] substitute the values in and calculate

b) Given a proportion, use the equation [pic] calculate p and q

c) To determine a sample size use [pic]

d) To determine a sample size given the proportions, use [pic]

Chapter 8 Practice

1. Suppose the commuting time from Georgetown to Washington varies uniformly from 20 to 45 min, depending on traffic and weather conditions. Construct a graph of this distribution and use the graph to find

a) The probability that a trip takes less than 30 min.

b) The probability that a trip takes more than 40 min.

2. A cookie manufacturer is randomly testing the diameter size of his companies’ cookies. He finds that the distribution of cookies is approximately normal with a mean diameter of 9.6 cm and a standard deviation of 1.13 cm. A cookie is rejected if it is too big (in the top 10%). What is the probability that

a) A randomly selected cookie has a diameter greater than 11 cm.

b) A randomly selected cookie has a diameter between 8.8 and 10.3 cm.

c) Will a cookie be rejected if it has a diameter of 11.8 cm?

3. Adrian’s average bowling score is 174, with a standard deviation of 35.

a) In what percent of games does Adrian score less than 200 points? At least 200 points?

b) The top 10% of bowlers in Adrian’s league get to play in an all-star game. If the league average is 170, with a standard deviation of 11 points, what average score does Arian need to have to obtain a spot in the all-star game?

4. The masses of statistics students are believed to be normally distributed. The masses (in kilograms) of a random sample of 36 statistics students are

|62 |63 |63 |64 |64 |65 |

|67 |67 |68 |68 |68 |68 |

|66 |67 |67 |67 |68 |68 |

|70 |70 |70 |71 |71 |71 |

|70 |71 |71 |71 |71 |72 |

|74 |75 |75 |77 |77 |78 |

a) Determine the mean and standard deviation of these data.

b) What is the probability that the mass of a randomly selected statistics student will be at least 70 kg?

c) Of 120 statistics students, how many would you expect to have a mass greater than 65 kg?

5. A multiple choice exam contains 50 questions and each question has 4 answers from which to choose. If a student merely guesses at the answers, what is the probability that

a) The student will get 10 questions correct

b) The student will pass

6. A machine produces articles and an average of 2% of theses articles are defective. In a batch of 400 articles what is the probability that no more than 4 are defective.

7. The theoretical probability of winning at the dice games, craps, is 0.493. If you play 50 games in Las Vegas, what is the probability that you will win more than you lose?

8. The fuel consumption for a new model of automobile is normally distributed with a mean of 30 km/L and a standard deviation of 3 km/L. Each week, the manufacturer asks 100 randomly selected owners of this new model to keep track of their fuel consumption.

a) Find the mean and standard deviation of the means of these samples.

b) How likely is the manufacturer to find a sample mean of 29 km/L or less?

9. A major soft drink company is interested in determining whether the company has made a significant improvement in its market share after a year long advertising campaign. The company has had a 24% share of the market historically. In a random sample of 500 customers, there are 152 of its products. Test whether there is a statistically significant increase in its market share at a significance level of 5%.

10. In a manufacturing process it has been found that 99.5% of the products are defect-free. A new process is introduced to improve the quality of the product. If in a batch of 3000, 2993 are defect free, has the new process significantly improved the quality of the product. Make a hypothesis test at a significance level of 10%.

11. A poll of 200 residents found that 73.0% support a new plan to revitalize a city’s waterfront. Determine a 95% confidence interval for support for the revitalization plan.

12. A company produces bolts with a nominal diameter of 10 mm. The diameters are normally distributed with a standard deviation of 0.1 mm. How large a sample would you need to test in order to be 99% confident of determining the actual mean diameter to within [pic]0.03 mm?

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download