Q1. Consider the random variable X, the number of runs conducted to produce an unacceptable lot when coating steel tubes. X is geometric with p = 0.05. Divide the 100 possible two-digit numbers into two categories, with numbers 00-04 denoting the production of an unacceptable lot and the remaining numbers denoting the production of an acceptable lot. Simulate the experiment of producing lots until an unacceptable one of obtained 10 times. Record the value obtained for X in each simulation. Based on these data, approximate the average value of X. Does your approximate value lie close to the theoretical mean value of 20? If not, run the simulation 10 more times. Is the arithmetic average of your observed values for X closer to 20 this time?
Q2. Verify the normal probability rule.
Q3. The average number of jets either arriving at or departing from O'Hare Airport is one every 40 seconds. What is the approxiamte probability that at least 75 such flights will occur during a randomly selected hour? What is the probability that fewer than 100 such flights will take place in an hour?
Q4. A satellite was malfunctioned and is expected to reenter the earth's atmosphere sometime during a 4-hour period. Let X denote the time of reentry. Assume that X is uniformly distributed over the interval [0, 4]. Simulate 20 observations on X.
Q5. A data set containing 70 observations, each reported to one decimal place, is to be split into seven categories. The largest observation is 75.1, and the smallest is 16.3
a) These data was covered by an interval of what length?
b) Using the method outlined in this section, each category will be of what length?
c) What is the lower boundary for the first category?
d) What are the boundaries for each of the seven categories?
Q6. Let X denote the time in minutes that a vehicle must wait to get through a traffic light at a busy intersection. The following data are obtained from a random sample of 36 vehicles:
0.2 0.5 0.7 1.1 1.2 1.3 1.4 1.4 1.4 1.5 1.5 1.6 1.6 1.7 1.9 2.0 2.1 2.1 2.2 2.3 2.5 2.6 2.9 2.8 3.0 3.1 3.0 3.7 3.7 4.0 4.1 4.5 5.1 5.8 1.4
a) Construct a double stem-and-leaf diagram for these data.
b) Do the data suggest that the distribution of X is skewed? If so, what is the direction of the skew?
Q7. Consider these data sets:
I II
1 3 2 1 2 4 1
2 5 4 2 5 2 5
4 3 3 1 5 5 3
a) Find the sample mean and sample median for each data set.
b) Find the sample range for each data set.
c) Find the sample variance and sample standard deviation for each data set.
d) Would you be surprised to hear someone claim that these data were drawn from the same poluation? Explain. Hint: Consider the shape of the distribution as well as the observed values of the sample statistics.
Q8. Consider the data of Q6
a) Find the mean and median for these data.
b) Find the standard deviation and variance for these data.
c) What physical measurement unit is associated with each of the statistics in parts a) and b)
Q9. (Approximating σ via the range.) The range can play an important role in the design of statistical studies. To obtain a prespecified degree of accuracy when estimating population parameters, an adequate sized sample must be drawn. Most formulas used to determine sample size require knowledge of σ, the population standard deviation. Often the researcher will not have an extimate of σ available but will have an idea of the expected range of his or her data. We saw that when sampling from a normal distribution, P[ -2σ < X - mean < 2σ ] = 0.95. If X is not normally distributed, then Chebyshev's inequality can be applied to conclude that P[ -3σ < X - mean < 3σ ] >= 0.89. That is, X always lies within at most 3 standard deviations of its mean with high probability. From this it can be concluded that be estimated range covers an interval of roughly 4σ for normally distributed random variables and 6σ otherwise. In the normal case an estimate of σ can be obtained by solving the equation 4σ = estimated range for σ. Thus we see that σ = (estimated range)/4 when X is normally distributed. If X is not noramlly distributed, then σ = (estimated range)/6. These data are obtained on the random variable X, the cpu time in seconds required to run a program using a statistical package: 6.2 5.8 4.6 4.9 7.1 5.2 8.1 0.2 3.4 4.5 8.0 7.9 6.1 5.6 5.5 3.1 6.8 4.6 3.8 2.6 4.5 4.6 7.7 3.8 4.1 6.1 4.1 4.4 5.2 1.5
a) Construct a stem-and-leaf diagram for these data. Is the assumption justified that X is normally distributed?
b) Approxiamte σ via the sample standard deviation s.
c) Find the sample range for these data, and use it to approximate σ. Compare your result to that obtained in part (b).
Q10. Let X be normally distributed when mean and variance.
a) Verify that q3 = mean + 0.67σ and that q1 = mean - 0.67σ.
b) Find the interquartile range for X.
c) Verify that the inner fences for X are f1 = mean - 2.68σ and f3 = mean + 2.68σ.
d) Verify that the probability that X will fall beyond the inner fences is approximately 0.007.
Q11. Let X denote the gasoline mileage obtained in tests on a newly designed SUV(sport utility vehicle). A sample of 21 simulated test runs yields these data:
15 16 17 18 17 20 16 17 18 20 18 18 19 19 17 21 17 19 18 17 22
a) Construct a stem-and-leaf diagram for these data. Do the data suggest that X is normally distributed?
b) Calculate the mean and median for this sample.
c) Calculate the standard deviation and variance for this sample.
d) Find the values of q1 and q3 and the iqr for the sample. Compare these values to these obtained via a TI83 calculator or any other technology tool that you have at your disposal.
Q12. Let X1, X2, X3, ...., X20 be a random sample from a distribution with mean 8 and variance 5. Find the mean and variance of X-bar.
Q13. Let X1, X2, X3, X4, X5 be a random sample from a binomial distribution with n = 10 and p unknown.
a) Show that X-bar/10 is an unbiased estimator for p.
b) Estimate p based on these data: 3, 4, 4, 5, 6
Q14. Let X1, X2, ..., Xn be a random saple from a Poisson distribution with parameter λs. Find the method of moments estimator for λs. Find the method of moments estimator for λ, the parameter underlying the Poisson process under observation.
Q15. Carbon dioxide is an odorless, colorless gas that constitutes about 0.035% by volume of the atmosphere. It affects the heat balance by acting as a one-way screen. It lets in the sun's heat to varm the oceans and the land but blocks some of the infrared heat that is radiated from the earth. This reflected heat is absorbed into the lower stmosphere, producing a greenhouse effect which causes the earh's surface to become warmer than it would be otherwidse. Systematic measurements of CO2 began in 1957 with Charles D. Keeling monitoring at Mauna Loa in Hawaii,
a) Assume that these CO2 readings (in ppm) are obtained:
319 338 337 339 328 325 340 331 341 336 330 330 321 327 337 320 343 350 322 334 326 349 341 338 332 339 335 338 333 334
Construct a stem-and-leaf diagram for these data using 31, 32, 32, 33, 33, 34, 34, 35 as steams. Graph leaves 0-4 on the first of each repeated steam and leaves 5-9 on the other. Is it reasonable to assume that the CO2 level in the atmosphere is normally distributed? Explain.
b) Estimate mean and variance using the method of moments estimators.
c) Find an unbiased estimate for variance.
Q16. Based on the data on Q15, what are the maximum likelihood estimates for the mean and variance of the atmospheric CO2 level?
Q17. Let W be an exponential random variable with parameter B unknown. Find the maximum likelihood estimator for B based on a sample of size n. Does it differ from the method of moments estimator?
Q18. To estimate the proportion of defective microprocessor chips being produced by a particular maker, samples of five chips are selected at 10 randomly selected times during the day. These chips are inspected, and X, the number of defective chips in each batch of size 5, is recorded. Assume that X is binomially distributed with n = 5 and p unknown. Use these data to find the maximum likelihood estimate for p:
1 0 1 2 0
0 0 0 1 0