Suppose that the weight of male babies less than 2 months old is normally distributed with mean 11.5 pounds and standard deviation 2.7 pounds. A sample of 36 babies is selected. What is the probability that the average weight of the sample is less than 11.07 pounds?
Write only a number as your answer. Round to 4 decimal places (for example 0.0048).
To find the probability that the average weight of the sample is less than 11.07 pounds, we can use the Central Limit Theorem. According to the Central Limit Theorem, when the sample size is large enough, the sampling distribution of the sample mean will be approximately normally distributed, regardless of the shape of the population distribution.
The mean of the sample mean can be calculated using the formula:
mean of sample mean = mean of population = 11.5 pounds
The standard deviation of the sample mean (also known as the standard error) can be calculated using the formula:
standard deviation of sample mean = standard deviation of population / √(sample size)
standard deviation of sample mean = 2.7 pounds / √36
standard deviation of sample mean = 2.7 pounds / 6
standard deviation of sample mean = 0.45 pounds
Now, we can standardize the value of 11.07 pounds using the z-score formula:
z = (x - mean) / standard deviation
z = (11.07 - 11.5) / 0.45
Calculating this, we get:
z = -0.93
Finally, we can find the probability using the standard normal distribution table. The table gives us the probability to the left of the z-score. In this case, we want to find the probability to the left of -0.93.
Looking up this value in the standard normal distribution table, we find that the probability is approximately 0.1762.
Therefore, the probability that the average weight of the sample is less than 11.07 pounds is approximately 0.1762.