The average age for licensed drivers in the county

is μ � 40.3 years with a standard deviation of � �
13.2 years.
a. A researcher obtained a random sample of n � 16
parking tickets and computed an average age of
M � 38.9 years for the drivers. Compute the z-score
for the sample mean and find the probability of
obtaining an average age this young or younger
for a random sample of licensed drivers. Is it
reasonable to conclude that this set of n � 16
people is a representative sample of licensed
drivers?
b. The same researcher obtained a random sample of
n � 36 speeding tickets and computed an average
age of M � 36.2 years for the drivers. Compute the
z-score for the sample mean and find the probability
of obtaining an average age this young or
younger for a random sample of licensed drivers.
Is it reasonable to conclude that this set of n � 36
people is a representative sample of licensed
drivers?

Z = (mean1 - mean2)/standard error (SE) of difference between means

SEdiff = √(SEmean1^2 + SEmean2^2)

SEm = SD/√n

If only one SD is provided, you can use just that to determine SEdiff.

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability of the Z scores.

To answer these questions, we need to use the z-score formula and find the probability using the standard normal distribution.

a. To calculate the z-score for the sample mean of 38.9 years, we use the formula:

z = (M - μ) / (σ / √n)

Where:
M = sample mean = 38.9 years
μ = population mean = 40.3 years
σ = standard deviation = 13.2 years
n = sample size = 16

Plugging in the values:

z = (38.9 - 40.3) / (13.2 / √16)
= -1.4 / (13.2 / 4)
= -1.4 / 3.3
= -0.424

Next, we need to find the probability of obtaining an average age this young or younger for a random sample of licensed drivers. This corresponds to finding the area under the standard normal distribution curve to the left of -0.424.

Using a standard normal distribution table or calculator, we can find that the probability is approximately 0.3365, or 33.65%.

Since this probability is not extremely low, it suggests that obtaining an average age of 38.9 years for a sample of 16 drivers is within the realm of possibility. Therefore, it is reasonable to conclude that this set of 16 people is a representative sample of licensed drivers.

b. Similarly, to calculate the z-score for the sample mean of 36.2 years, we use the same formula:

z = (M - μ) / (σ / √n)

Using the given values:

z = (36.2 - 40.3) / (13.2 / √36)
= -4.1 / (13.2 / 6)
= -4.1 / 2.2
= -1.864

Next, we find the probability of obtaining an average age this young or younger for a random sample of licensed drivers, which corresponds to the area under the standard normal distribution curve to the left of -1.864.

Using the standard normal distribution table or calculator, we can find that the probability is approximately 0.0310, or 3.10%.

Since this probability is quite low, it suggests that obtaining an average age of 36.2 years for a sample of 36 drivers is unlikely to happen by chance. Therefore, it is not reasonable to conclude that this set of 36 people is a representative sample of licensed drivers.

To solve these questions, we need to apply the concepts of z-scores and the normal distribution.

a) To compute the z-score for the sample mean of 38.9 years, we use the formula:

z = (x - μ) / (σ / sqrt(n))

where x is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.

Given:
Population mean (μ) = 40.3 years
Population standard deviation (σ) = 13.2 years
Sample mean (x) = 38.9 years
Sample size (n) = 16

Plugging in the values, we get:

z = (38.9 - 40.3) / (13.2 / sqrt(16))

Simplifying,

z = -1.4 / (13.2 / 4)

z = -1.4 / 3.3

z ≈ -0.4242

To find the probability of obtaining an average age this young or younger, we need to look up the z-score in the z-table.

The z-table gives the area under the standard normal curve to the left of a given z-score. In this case, we want to find the probability to the left of z = -0.4242.

Looking up the z-score in the z-table, we find that the corresponding probability is approximately 0.3357.

Therefore, the probability of obtaining an average age this young or younger for a random sample of licensed drivers is approximately 0.3357 or 33.57%.

To determine if this set of n = 16 people is a representative sample of licensed drivers, we need additional information. The probability alone does not provide sufficient evidence. Other factors such as sampling method, representativeness, and randomness should be considered.

b) To compute the z-score for the sample mean of 36.2 years, we follow the same steps as in part (a):

Given:
Population mean (μ) = 40.3 years
Population standard deviation (σ) = 13.2 years
Sample mean (x) = 36.2 years
Sample size (n) = 36

z = (36.2 - 40.3) / (13.2 / sqrt(36))

Simplifying,

z = -4.1 / (13.2 / 6)

z = -4.1 / 2.2

z ≈ -1.8636

To find the probability of obtaining an average age this young or younger, we look up the z-score in the z-table.

The z-table gives the area under the standard normal curve to the left of a given z-score. In this case, we want to find the probability to the left of z = -1.8636.

Looking up the z-score in the z-table, we find that the corresponding probability is approximately 0.0312.

Therefore, the probability of obtaining an average age this young or younger for a random sample of licensed drivers is approximately 0.0312 or 3.12%.

Similar to part (a), determining if this set of n = 36 people is a representative sample of licensed drivers would require additional information beyond just the probability calculated.