The average life expectancy of Canadian women is 75 years, and is normally distributed. The standard deviation is known to be 7 years. If a random sample od 49 women is taken, what is the probability that the average age of the sample of women is:
Between 73.5 and 76 years?
Between 72 and 74 years?
Less than 72.7 years?
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z scores.
Z = (score-mean)/SEm
SEm = SD/√n
73.5-76=2.5/7 = 0.357
To find the probabilities for the given scenarios, we will use the concept of the sampling distribution of the sample mean. The mean of the sampling distribution will be the same as the population mean, which is 75 years in this case. The standard deviation of the sampling distribution, also known as the standard error, can be calculated by dividing the population standard deviation by the square root of the sample size, which is √49 = 7.
Let's calculate the probabilities for each scenario:
1. Between 73.5 and 76 years:
To find this probability, we need to calculate the z-scores for the lower and upper values. The z-score formula is (x - μ) / σ, where x is the value, μ is the mean, and σ is the standard deviation.
For the lower value (73.5):
z1 = (73.5 - 75) / 7
For the upper value (76):
z2 = (76 - 75) / 7
Next, we need to find the areas under the normal curve for these z-scores. We can do this by using a standard normal distribution table or a calculator such as Excel or an online tool.
The probability between 73.5 and 76 years is calculated as:
P(73.5 < X < 76) = P(z1 < Z < z2)
2. Between 72 and 74 years:
Similarly, we calculate the z-scores for the lower and upper values of this range:
For the lower value (72):
z1 = (72 - 75) / 7
For the upper value (74):
z2 = (74 - 75) / 7
The probability between 72 and 74 years is calculated as:
P(72 < X < 74) = P(z1 < Z < z2)
3. Less than 72.7 years:
To find this probability, we calculate the z-score for the given value (72.7):
z = (72.7 - 75) / 7
The probability of being less than 72.7 years is calculated as:
P(X < 72.7) = P(Z < z)
To find the probabilities, you can use a standard normal distribution table or a calculator that can calculate probabilities from the normal distribution, such as Excel or an online tool.