Listed Below Are The 35 Members Of The Metro Toledo Automobi

Listed Below Are The 35 Members Of the Metro Toledo Automobile Dealers

We would like to estimate the mean revenue from dealer service departments among the members of the Metro Toledo Automobile Dealers Association. There are 35 members, each identified by numbers 00 through 34. A random sample of five dealers is to be selected, with the sample numbers given as 5, 22, 43, 24, 34, 28, 46, 36, 68, 8, 30, and 2. The task is to determine which dealers are included in the sample based on these random numbers, considering only dealer numbers from 00 to 34, and to compute the probability that the sample has a mean revenue of at least $116,000. In addition, given a normal population with mean 77 and standard deviation 5, and a sample size of 48, the probability that the sample mean is less than 76 is to be calculated, along with the p-value.

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The primary objective of this analysis is to estimate the mean revenue from dealer service departments among the members of the Metro Toledo Automobile Dealers Association, a group comprising 35 members. These members are numbered from 00 through 34, and selecting a representative sample is crucial for making meaningful inferences about the population’s revenue characteristics. The specific task involves identifying the dealers included in a randomly selected sample based on provided random numbers, and subsequently calculating the probability associated with certain sample mean outcomes, assuming a normal distribution of revenues.

First, understanding how to select the sample involves matching the given random numbers to the dealer numbers, limiting the selection to numbers between 00 and 34. Of the list provided (5, 22, 43, 24, 34, 28, 46, 36, 68, 8, 30, 2), only those numbers within this range are valid sample points. Therefore, the dealers included in the sample are dealers numbered 5, 22, 24, 34, 28, 8, 30, and 2. This yields an 8-dealer subset from the initial random list for the sample.

Next, the probability calculation concerns the likelihood of selecting a sample where the mean revenue is at least $116,000. To compute this probability, various assumptions about the population distribution, sample size, and the standard deviation are necessary. Under typical statistical procedures, the probability that the sample mean exceeds a particular value can be determined using the standard normal distribution. The z-score is calculated by subtracting the population mean from the sample mean and dividing by the standard error (standard deviation divided by the square root of the sample size). In practice, if the population mean and standard deviation are known, the probability that the sample mean is at least $116,000 is given by P(ȳ ≥ $116,000) = 1 - P(Z ≤ z), where Z is the standard normal variable.

Additionally, considering the scenario where the population has a mean of 77 and a standard deviation of 5, and a sample size of 48, the probability that the sample mean is less than 76 is computed similarly. The z-value in this case is (76 - 77) / (5 / √48). Using this z-score, the corresponding probability is obtained from the standard normal distribution table. Furthermore, the p-value, which indicates the probability of observing a sample mean as extreme or more extreme under the null hypothesis, is estimated accordingly. This p-value assists in hypothesis testing to determine the significance of the observed sample mean.

In conclusion, selecting the correct sample dealers from the provided list involves filtering the random numbers within the dealer range, then calculating associated probabilities using standard techniques in inferential statistics. These calculations help in understanding the likelihood of certain revenue outcomes and support decision-making regarding dealer performance in the context of the dealership association. Such statistical insights are essential for strategic planning, resource allocation, and assessing operational efficiency of the dealer network.

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