Simulated data is frequently used in economic research. An example of how a simulated data set may be constructed is through using a random number generator. Consider when random numbers between 0 and 1 are generated, with each generated number having four decimal places. Now decide that a sample of 10,000 randomly generated numbers is to be selected. These 10,000 numbers will form what is called a data set. If the mean of distribution of the individual random numbers between 0 and 1 is 0.5 and the variance 1/12, calculate: a) The probability that a random sample of 10,000 numbers (a data set) will have a sample mean of at least 0.499 (Answer to 4 decimal places): b) Using the probability from part a, if 5,000 data sets include 10,000 random numbers, how many of these data sets would be expected to have a sample mean of at least 0.499 (calculate to the nearest whole number)? Answer:
Question
Simulated data is frequently used in economic research. An example of how a simulated data set may be constructed is through using a random number generator. Consider when random numbers between 0 and 1 are generated, with each generated number having four decimal places. Now decide that a sample of 10,000 randomly generated numbers is to be selected. These 10,000 numbers will form what is called a data set. If the mean of distribution of the individual random numbers between 0 and 1 is 0.5 and the variance 1/12, calculate: a) The probability that a random sample of 10,000 numbers (a data set) will have a sample mean of at least 0.499 (Answer to 4 decimal places): b) Using the probability from part a, if 5,000 data sets include 10,000 random numbers, how many of these data sets would be expected to have a sample mean of at least 0.499 (calculate to the nearest whole number)? Answer:
Solution 1
a) To find the probability that a random sample of 10,000 numbers will have a sample mean of at least 0.499, we first need to find the standard deviation of the mean. The standard deviation of the mean is the standard deviation divided by the square root of the number of observations (σ / √n), which in this case is √(1/12) / √10000 = 0.002887 (rounded to six decimal places).
We then standardize the value 0.499:
Z = (X - μ) / σ
where X is the value we are interested in (0.499 in this case), μ is the mean (0.5), and σ is the standard deviation of the mean (0.002887).
Z = (0.499 - 0.5) / 0.002887 = -0.346 (rounded to three decimal places)
We then look up this z-score in a standard normal distribution table or use a calculator with a normal distribution function to find the probability. The probability that Z is less than -0.346 is 0.3643 or 36.43%. However, we want the probability that the sample mean is at least 0.499, so we need to subtract this probability from 1: 1 - 0.3643 = 0.6357 or 63.57%.
b) If 5,000 data sets each include 10,000 random numbers, and each data set has a 63.57% chance of having a sample mean of at least 0.499, then we would expect approximately 0.6357 * 5000 = 3178.5 of these data sets to have a sample mean of at least 0.499. Rounding to the nearest whole number, we would expect approximately 3179 data sets to have a sample mean of at least 0.499.
Solution 2
No answer
Similar Questions
Random NumbersWhat is the use of the rnorm() function?It is used to generate random numbers from a normal distributionIt is used to generate random numbers from a uniform distributionIt is used to ensure that the same set of random numbers is reproduced every time that block of code is runNone of the above
Which technique for gathering data (sampling, experiment, simulation, or census) do you think was used in the following studies?(a) An analysis of a sample of 31,000 patients from New York hospitals suggests that the poor and the elderly sue for malpractice at one-fifth the rate of wealthier patients. (Journal of the American Medical Association).samplingexperiment simulationcensus(b) The effects of wind shear on airplanes during both landing and takeoff were studied by using complex computer programs that mimic actual flight.samplingexperiment simulationcensus(c) A study of all league football scores attained through touchdowns and field goals was conducted by the National Football League to determine whether field goals account for more scoring events than touchdowns (USA Today).samplingexperiment simulationcensus(d) An Australian study included 588 men and women who already had some precancerous skin lesions. Half got skin cream containing a sunscreen with a sun protection factor of 17; half got an inactive cream. After 7 months, those using the sunscreen with the sun protection had fewer precancerous skin lesions (New England Journal of Medicine).samplingexperiment simulationcensus
Compare the calculated data with the simulated results.____________________________________________________________________________________________________________________________________________
A researcher wants to study the eating habits of college students. They assign each student in the college a unique number and use a random number generator to select a sample of 100 students.What kind of sampling method is being used?Group of answer choicesStratified Random SamplingCluster SamplingSystematic SamplingRandom Sampling
Youexplain why we use random sampling technique on our thesis
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.