Question

the distribution of the number of transactions per day at a certain automated teller machine (atm) is approximately normal with a mean of 80 transactions and a standard deviation of 10 transactions. which of the following represents the parameters of the distribution?
a ( \bar{x} = 80; s = 10 )
b ( \bar{x} = 80; s^2 = 10 )
c ( \bar{x} = 80; sigma = 10 )
d ( mu = 80; sigma = 10 )
e ( mu = 80; s = 10 )

Explanation:

Step1: Recall Normal Distribution Parameters

For a normal distribution, the population mean is denoted by $\mu$ and the population standard deviation is denoted by $\sigma$. Sample statistics use $\bar{x}$ (sample mean) and $s$ (sample standard deviation), but here we are dealing with the distribution of the number of transactions (the population), so we use population parameters.
The mean of the distribution is 80, so $\mu = 80$. The standard deviation is 10, so $\sigma = 10$.

Step2: Analyze Each Option

• Option A: $\bar{x}$ (sample mean) and $s$ (sample standard deviation) are for samples, not the population distribution. Eliminate.
• Option B: $s^2$ is sample variance, and we need standard deviation, also using sample statistic. Eliminate.
• Option C: $\bar{x}$ is sample mean, not population mean. Eliminate.
• Option D: $\mu = 80$ (population mean) and $\sigma = 10$ (population standard deviation) match the normal distribution parameters.
• Option E: $s$ is sample standard deviation, not population. Eliminate.

Answer:

D. $\mu = 80$; $\sigma = 10$