The distribution of heights of American women aged 18 to 24 is approximately normally distributed with mean 65.5 inches and standard deviation 2.5 inches. What is the probability that a randomly selected woman is over 70 inches tall?
Z = (score-mean)/SD
Calculate the Z score and find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to that Z score.
To find the probability that a randomly selected woman is over 70 inches tall, we need to use the concept of z-scores and the standard normal distribution.
Step 1: Determine the z-score
The z-score is a measure of how many standard deviations an observation is away from the mean. In this case, we want to determine the z-score for 70 inches.
z-score = (x - mean) / standard deviation
z-score = (70 - 65.5) / 2.5
Step 2: Look up the z-score in the standard normal distribution table
Using the z-score from Step 1, locate the corresponding probability in the standard normal distribution table. The table will give us the probability of a z-score less than or equal to the given value, so we need to find the complement of that probability.
Step 3: Calculate the probability of being over 70 inches
Since we want the probability that the height is over 70 inches, we need to subtract the probability we found in Step 2 from 1.
To summarize:
Step 1: Calculate the z-score: z-score = (70 - 65.5) / 2.5
Step 2: Look up the probability from the standard normal distribution table.
Step 3: Calculate the probability of being over 70 inches: probability = 1 - probability from Step 2.
To find the probability that a randomly selected woman is over 70 inches tall, we need to calculate the area under the normal distribution curve to the right of 70 inches.
Step 1: Standardize the value
To standardize the value of 70 inches, we need to calculate the z-score. The z-score formula is:
z = (x - μ) / σ
where x is the value we want to standardize, μ is the mean, and σ is the standard deviation.
In this case, we want to standardize 70 inches using the mean of 65.5 inches and the standard deviation of 2.5 inches:
z = (70 - 65.5) / 2.5
z = 4.5 / 2.5
z = 1.8
Step 2: Look up the z-score
Next, we need to look up the area under the normal distribution curve corresponding to a z-score of 1.8. We can use a standard normal distribution table or a calculator to find this value. The area to the right of 1.8 is the probability of being over 70 inches tall.
Using a standard normal distribution table or calculator, we find that the area to the right of 1.8 is approximately 0.0359.
So, the probability that a randomly selected woman is over 70 inches tall is approximately 0.0359, or 3.59%.