Answer:
1: 0.8461 = 84.61% probability that a randomly selected 1-year old boy from the population will have a weight that is less than 25 lbs.
2: 0.9994 = 99.94% probability that the mean weight for a sample of size 10 1-year-old boys will be less than 25 lbs
3: In part 2, we use the sampling distribution of the sample means, which has more values closer to the mean due to the smaller standard error, so a higher probability of finding a mean less than 25 lbs.
Explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal Probability Distribution:
Problems of normal distributions can be solved using the z-score formula.
In a set with mean
and standard deviation
, the z-score of a measure X is given by:
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean
and standard deviation
, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
and standard deviation
.
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
Mean of 22.8 lbs and a standard deviation of about 2.15 lbs.
This means that
Part 1: Find the probability that a randomly selected 1-year old boy from the population will have a weight that is less than 25 lbs.
This is the pvalue of Z when X = 25. So
has a pvalue of 0.8461
0.8461 = 84.61% probability that a randomly selected 1-year old boy from the population will have a weight that is less than 25 lbs.
Part 2: Find the probability that the mean weight for a sample of size 10:
Now
. So
By the Central Limit Theorem
has a pvalue of 0.9994
0.9994 = 99.94% probability that the mean weight for a sample of size 10 1-year-old boys will be less than 25 lbs
Part 3: Explain the difference between Part 1 and Part 2.
In part 2, we use the sampling distribution of the sample means, which has more values closer to the mean due to the smaller standard error, so a higher probability of finding a mean less than 25 lbs.