Statistical Abstracts (117th edition) provides information aboutthe number of
registered automobiles, including taxis, by state. The number ofregistered
automobiles per 1,000 residents follows an approximately normaldistribution with
mean ì = 514.6 and standard deviation ó = 84.6. In thequestions below, x = number
of registered automobiles per 1,000 residents.
(a) Find the probability that for a random sample of 4 states thesample mean, x is
between 475 and 525.
(b) Find the probability that for a random sample of 16 states thesample mean, x ,
is between 475 and 525.
To find the probability in both parts (a) and (b), we need to use the concept of the sampling distribution of the sample mean. The central limit theorem states that for a sufficiently large sample size, the sampling distribution of the sample mean will be approximately normally distributed, regardless of the shape of the population distribution.
For part (a):
Step 1: Calculate the standard error of the sample mean
The standard error of the sample mean (σx̄) is given by the formula:
σx̄ = σ / √n
where σ is the standard deviation of the population (84.6) and n is the sample size (4).
σx̄ = 84.6 / √4
σx̄ = 84.6 / 2
σx̄ = 42.3
Step 2: Standardize the values 475 and 525 using the sample mean standard error
To standardize a value x, we subtract the mean (514.6) and divide by the standard error (42.3).
For 475:
Z1 = (475 - 514.6) / 42.3
Z1 = -39.6 / 42.3
Z1 = -0.936
For 525:
Z2 = (525 - 514.6) / 42.3
Z2 = 10.4 / 42.3
Z2 = 0.246
Step 3: Find the probability using the standard normal distribution table
Using the standard normal distribution table or a calculator, find the area under the curve between Z1 and Z2. This represents the probability that the sample mean (x) is between 475 and 525.
The probability can be calculated as follows:
P(475 < x < 525) = P(-0.936 < Z < 0.246)
For part (b):
The steps are similar to part (a), but we need to adjust the standard error since the sample size is different.
Step 1: Calculate the standard error of the sample mean
The standard error (σx̄) is given by the formula:
σx̄ = σ / √n
where σ is the standard deviation of the population (84.6) and n is the sample size (16).
σx̄ = 84.6 / √16
σx̄ = 84.6 / 4
σx̄ = 21.15
Step 2: Standardize the values 475 and 525 using the sample mean standard error
To standardize a value x, we subtract the mean (514.6) and divide by the standard error (21.15).
For 475:
Z1 = (475 - 514.6) / 21.15
Z1 = -39.6 / 21.15
Z1 = -1.876
For 525:
Z2 = (525 - 514.6) / 21.15
Z2 = 10.4 / 21.15
Z2 = 0.492
Step 3: Find the probability using the standard normal distribution table
Using the standard normal distribution table or a calculator, find the area under the curve between Z1 and Z2. This represents the probability that the sample mean (x) is between 475 and 525.
The probability can be calculated as follows:
P(475 < x < 525) = P(-1.876 < Z < 0.492)
Note: If you have access to statistical software like R or Excel, you can also use these tools to calculate the probability directly using the normal distribution functions.
Use a z-table and z-scores. You will have to account for the sample size, so here is the formula:
z = (x - mean)/(sd/√n)
For a): calculate two z-scores
z = (475 - 514.6)/(84.6/√4)
z = (525 - 514.6)/(84.6/√4)
Once you have the two z-scores, find the probability between the two using a z-table.
For b): calculate two z-scores using the same data as a), with the exception of 16 for the sample size.
Find the probability between the two z-scores using a z-table.
I hope this will help get you started.