Answer:
1. P(x>107,2) = 0,5119
2. P(x(bar)> 107,2) = 0,6772
Step-by-step explanation:
Hello!
Data:
The population of values with normal distribution
μ: 107,7
σ: 16,8
n=243
1. According to the text: X≈N(μ; σ2)
To standardize it and calculate the asked probability we can use is
Z= (x-μ)/σ≈N(0;1)
First, let's rewrite the probability to its complement since most of the tables of probability accumulate from left to right
P(X>107,2)= 1- P(X≤107,2)
Then we standardize
1-P(Z≤(x-μ)/σ) = 1- P(Z≤((107,2-107,7)/16,8) = 1- P(Z≤-0,029) = 1- P(Z≤-0,03)
= 1- 0,48803 = 0,51197
2. The next probability asked is not about a random value from the sample (x) but for a value that the sample mean might take. To calculate this probability we need to take the distribution of the sample mean into consideration.
This is x(bar) ≈ N(μ;δ2/n)
For standardization, we will also use the Z distribution, but under the distribution of the sample mean. Since the mean is for the same population, the values that μ and δ take are the same, but in this case, the sample n also plays a role in the formula.
In this case the statistic will be Z= (x(bar) - μ)/ δ/√n ≈N(0;1)
For the probability
P(x(bar) > 107,2) = 1 - P(x(bar)≤107,2)
1 - P(Z ≤ (x(bar) - μ)/ δ/√n) = 1 - P(Z ≤ (107,2 - 107,7)/ 16,8/√243) = 1 - P(Z ≤ -0,46)
1 - P(Z ≤ -0,46) = 1 - 0,32276 = 0,67724
I hope you have a SUPER day!
