SECTION 5.4 Sampling Distributions and the Central Limit Theorem 273 Extending Concepts Finite Correction Factor The formula for the standard deviation of the sampling distribution of sample means sx = s2 n given in the Central Limit Theorem is based on an assumption that the population has infinitely many members. This is the case whenever sampling is done with replacement (each member is put back after it is selected), because the sampling process could be continued indefinitely. The formula is also valid when the sample size is small in comparison with the population. When sampling is done without replacement and the sample size n is more than 5% of the finite population of size N 1n N 7 0.052, however, there is a finite number of possible samples. A finite correction factor, AN - n N - 1 should be used to adjust the standard deviation. The sampling distribution of the sample means will be normal with a mean equal to the population mean, and the standard deviation will be sx = s1 nAN - n N - 1 . In Exercises 39 and 40, determine whether the finite correction factor should be used. If so, use it in your calculations when you find the probability. 39. Parking Infractions In a sample of 1000 fines issued by the City of Toronto for parking infractions in September of 2020, the mean fine was $49.83 and the standard deviation was $52.15. A random sample of size 60 is selected from this population. What is the probability that the mean fine is less than $40? (Adapted from City of Toronto) 40. Old Faithful In a sample of 100 eruptions of the Old Faithful geyser at Yellowstone National Park, the mean interval between eruptions was 129.58 minutes and the standard deviation was 108.54 minutes. A random sample of size 30 is selected from this population. What is the probability that the mean interval between eruptions is between 120 minutes and 140 minutes? (Adapted from Geyser Times) Sampling Distribution of Sample Proportions For a random sample of size n, the sample proportion is the number of individuals in the sample with a specified characteristic divided by the sample size. The sampling distribution of sample proportions is the distribution formed when sample proportions of size n are repeatedly taken from a population where the probability of an individual with a specified characteristic is p. The sampling distribution of sample proportions has a mean equal to the population proportion p and a standard deviation equal to 2pq n. In Exercises 41 and 42, assume the sampling distribution of sample proportions is a normal distribution. 41. Construction About 63% of the residents in a town are in favor of building a new high school. One hundred five residents are randomly selected. What is the probability that the sample proportion in favor of building a new school is less than 55%? Interpret your result. 42. Conservation About 74% of the residents in a town say that they are making an effort to conserve water or electricity. One hundred ten residents are randomly selected. What is the probability that the sample proportion making an effort to conserve water or electricity is greater than 80%? Interpret your result.
RkJQdWJsaXNoZXIy NjM5ODQ=