The average score of all golfers for a particular course has a mean of 73 and a standard deviation of 3. Suppose
36 golfers played the course today. Find the probability that the average score of the 36 golfers exceeded 74.
To find the probability that the average score of the 36 golfers exceeded 74, we need to use the concept of sampling distributions and the Central Limit Theorem.
Here are the steps to solve this problem:
Step 1: Define the variables
Let X be the individual score of a golfer. We know that the mean score (μ) is 73 and the standard deviation (σ) is 3.
Step 2: Define the sample size
In this case, the sample size (n) is 36, representing the number of golfers who played the course today.
Step 3: Define the sample mean
The sample mean (X-bar) is the average score of the 36 golfers.
Step 4: Understand the Central Limit Theorem
According to the Central Limit Theorem, when the sample size is sufficiently large (n > 30), the distribution of the sample means approaches a normal distribution, regardless of the shape of the original population distribution.
Step 5: Calculate the standard error (SE)
The standard error (SE) is the standard deviation of the sample mean, which is calculated using the formula: SE = σ / sqrt(n). In this case, σ (standard deviation) is 3, and n (sample size) is 36. Therefore, SE = 3 / sqrt(36), which equals 0.5.
Step 6: Convert X-bar to a standard score (Z-score)
To find the probability, we need to convert the average score of the 36 golfers (X-bar) to a standard score (Z-score), which is done using the formula: Z = (X-bar - μ) / SE. In this case, X-bar is 74 (as we want to find the probability that it exceeds 74), μ (mean) is 73, and SE (standard error) is 0.5. Therefore, Z = (74 - 73) / 0.5 = 2.
Step 7: Find the probability using the Z-table
Look up the Z-score of 2 in the Z-table, which represents the area under the standard normal distribution curve to the left of a given Z-score. The Z-table will give you the corresponding cumulative probability.
The probability that the average score of the 36 golfers exceeded 74 is the complement of the cumulative probability found in the Z-table. Subtract the cumulative probability from 1 to get the probability that exceeds 74.
Note: Since the Z-table provides the probabilities for a Z-score to the left of a specific value, we need to find the area to the left of Z = 2 and subtract it from 1 to find the area to the right.
So, using the Z-table, the probability that the average score of the 36 golfers exceeded 74 can be found by subtracting the cumulative probability (from Z = -∞ to Z = 2) from 1.
To find the probability that the average score of the 36 golfers exceeded 74, we need to calculate the z-score and use the standard normal distribution.
Step 1: Calculate the standard error of the mean
The standard error of the mean (SE) is calculated by dividing the standard deviation by the square root of the sample size:
SE = 3 / √36
SE = 3 / 6
SE = 0.5
Step 2: Calculate the z-score
The z-score is calculated using the formula:
z = (x - μ) / SE
where x is the desired value (74 in this case), μ is the mean (73), and SE is the standard error.
z = (74 - 73) / 0.5
z = 1 / 0.5
z = 2
Step 3: Look up the z-score in the z-table
Using a z-table (or calculator), we can find the cumulative probability associated with a z-score of 2. The cumulative probability for a z-score of 2 is approximately 0.9772.
Step 4: Subtract the cumulative probability from 1
Since we want to find the probability that the average score exceeded 74, we need to subtract the cumulative probability from 1:
P(x > 74) = 1 - 0.9772
P(x > 74) = 0.0228
Therefore, the probability that the average score of the 36 golfers exceeded 74 is approximately 0.0228 or 2.28%.
Z = (mean1 - mean2)/SEm
SEm = SD/√n
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to the Z score.