The lengths of pregnancies are normally distributed with a mean of 273 days and a standard deviation of 20 days. If 64 women are randomly selected, find the probability that they have a mean pregnancy between 273 days and 275 days.
2/2.5 = 0.8 z=0.7881 1-0.7881 = 0.2119
Well, this question is certainly expecting a lot from me! But don't worry, I'm here to deliver the probability to you with a smile!
To find this probability, we need to use the concept of the Central Limit Theorem. According to this theorem, when we have a sample size that is large enough (and 64 is definitely large enough), the distribution of the sample means will be approximately normally distributed, regardless of the shape of the original population.
First, let's calculate the z-scores for the given values. The z-score formula is:
z = (x - μ) / (σ / √n)
For the lower bound, 273 days, we have:
z1 = (273 - 273) / (20 / √64) = 0
For the upper bound, 275 days, we have:
z2 = (275 - 273) / (20 / √64) = 2 / (20 / √64) ≈ 0.8
Now we need to find the probability associated with these z-scores using a standard normal distribution table (or a calculator). The probability associated with a z-score of 0 is 0.5, and the probability associated with a z-score of 0.8 is approximately 0.7881.
So, to find the probability that the mean pregnancy will be between 273 and 275 days, we subtract the probability associated with the lower z-score from the probability associated with the upper z-score:
P(273 ≤ x ≤ 275) = P(0 ≤ z ≤ 0.8) = 0.7881 - 0.5 = 0.2881
Therefore, the probability that the mean pregnancy will be between 273 and 275 days for these 64 women is approximately 0.2881, or about 28.81%.
I hope this answer brightened up your day a little!
To find the probability that the mean pregnancy of 64 women falls between 273 and 275 days, we need to calculate the z-scores for both values and then find the corresponding probabilities using the standard normal distribution.
First, let's calculate the z-score for a mean pregnancy of 273 days:
z = (X - μ) / (σ / sqrt(n))
= (273 - 273) / (20 / sqrt(64))
= 0 / (20 / 8)
= 0
Next, let's calculate the z-score for a mean pregnancy of 275 days:
z = (X - μ) / (σ / sqrt(n))
= (275 - 273) / (20 / sqrt(64))
= 2 / (20 / 8)
= 2 / 2
= 1
Now, we can find the probabilities associated with these z-scores using a standard normal distribution table or a calculator.
The probability that the mean pregnancy is between 273 and 275 days can be calculated as follows:
P(273 ≤ X ≤ 275) = P(0 ≤ Z ≤ 1)
Using a standard normal distribution table, the probability corresponding to a z-score of 0 is 0.5000, and the probability corresponding to a z-score of 1 is 0.8413.
Therefore, the probability that the mean pregnancy of 64 women falls between 273 and 275 days is:
P(273 ≤ X ≤ 275) = P(0 ≤ Z ≤ 1)
= P(Z ≤ 1) - P(Z ≤ 0)
= 0.8413 - 0.5000
= 0.3413
So, the probability is approximately 0.3413 or 34.13%.
To find the probability that the mean pregnancy length of the 64 selected women is between 273 days and 275 days, we first need to standardize the distribution.
The mean of the distribution is 273 days, and the standard deviation is 20 days. We want to find the probability of the mean being between 273 days and 275 days.
To standardize a distribution, we use the formula:
z = (x - μ) / σ
where z is the standardized score, x is the given value, μ is the mean, and σ is the standard deviation.
Now we can calculate the standardized scores for both 273 and 275 days:
For 273 days:
z1 = (273 - 273) / 20 = 0
For 275 days:
z2 = (275 - 273) / 20 = 0.1
Next, we can find the probability associated with these standardized scores using the standard normal distribution table or a statistical calculator.
Using the standard normal distribution table, we find that the area to the left of z1 (0) is 0.5000 and the area to the left of z2 (0.1) is 0.5398.
To find the probability between these two values, we subtract the area to the left of z1 from the area to the left of z2:
P(273 ≤ X ≤ 275) = P(z1 ≤ Z ≤ z2) = P(0 ≤ Z ≤ 0.1) = 0.5398 - 0.5000 = 0.0398
Therefore, the probability that the mean pregnancy length of the 64 selected women is between 273 days and 275 days is approximately 0.0398, or 3.98%.
Use z-scores:
z = (275 - 273)/(20/√64) = ?
Once you calculate the z-score, check a z-table from mean to that z for your probability.