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.


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?

Verified step by step guidance

1. Step 1: Determine whether the finite population correction factor (FPC) should be used. The FPC is applied when the sample size (n) is large relative to the population size (N). Specifically, it is used if n > 0.05N. In this case, the population size is 1000, and the sample size is 60. Calculate 0.05N = 0.05 × 1000 = 50. Since n = 60 > 50, the FPC should be used.
2. Step 2: Calculate the finite population correction factor using the formula: \( \text{FPC} = \sqrt{\frac{N - n}{N - 1}} \), where N is the population size and n is the sample size. Substitute N = 1000 and n = 60 into the formula.
3. Step 3: Adjust the standard error of the mean using the FPC. The formula for the adjusted standard error is: \( \text{Adjusted SE} = \text{SE} \times \text{FPC} \), where \( \text{SE} = \frac{\sigma}{\sqrt{n}} \) and \( \sigma \) is the population standard deviation. First, calculate \( \text{SE} \) using \( \sigma = 52.15 \) and \( n = 60 \), then multiply by the FPC calculated in Step 2.
4. Step 4: Standardize the sample mean to find the z-score. Use the formula: \( z = \frac{\bar{x} - \mu}{\text{Adjusted SE}} \), where \( \bar{x} \) is the sample mean, \( \mu \) is the population mean, and \( \text{Adjusted SE} \) is the adjusted standard error from Step 3. Substitute \( \bar{x} = 40 \), \( \mu = 49.83 \), and the adjusted standard error.
5. Step 5: Use the z-score to find the probability. Look up the z-score in a standard normal distribution table or use statistical software to find the cumulative probability corresponding to the z-score. This cumulative probability represents the probability that the mean fine is less than \$40.

Key Concepts

Finite Correction Factor

The Finite Correction Factor (FCF) is used in statistics when sampling without replacement from a finite population. It adjusts the standard error of the sample mean to account for the fact that the sample size is a significant fraction of the total population. When the sample size is more than 5% of the population, the FCF is applied to ensure more accurate probability calculations.

Sampling Distribution of the Mean

The Sampling Distribution of the Mean describes the distribution of sample means from a population. According to the Central Limit Theorem, as the sample size increases, the distribution of the sample means approaches a normal distribution, regardless of the population's shape. This concept is crucial for calculating probabilities related to sample means, especially when determining how likely it is for the sample mean to fall below a certain value.

Z-Score

A Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It is calculated by subtracting the mean from the value and then dividing by the standard deviation. Z-scores are essential for determining probabilities in a normal distribution, as they allow us to find the likelihood of a sample mean being less than a specific value by referencing standard normal distribution tables.