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.
36 golfers played the course today. Find the probability that the average score of the 36 golfers exceeded 74.
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.
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.
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%.