In a normal distribution, 4% of the distribution is less than 53 and 97% of the distribution is less than 65.i.Find the mean and standard deviation of the distribution. ii.Find the interquartile range of the distribution.
I have now answered several of these same type of problems for you.
This time show me your steps and your conclusion so I can take a look at it
hint:
What z-score corresponds with P(x < 53) = .04 ?
etc
P(X<53)=0.04
P(X<65)=0.97
53=m-1.751sd----1)
65=m-1.882sd-----2)
Solving this didn't gave me 58.79 which is the answer.
less than 4% ----> z-score = -1.751
less than 97% ---> z-score = 1.881
(53-m)/s = -1.751 ---> m - 1.751s = 53
(65-m)/s = 1.881s = 65 ---> m + 1.881s = 65
subtract them:
3.632s = 12
s = 3.304, then m = 58.785
your 2nd equation was wrong
Can you look the previous question where I added one more question?
To find the mean and standard deviation of the distribution, we can use the properties of the standard normal distribution.
i. Finding the mean and standard deviation:
1. Convert the given percentage values to z-scores using the standard normal distribution table or a calculator. The z-score represents how many standard deviations away from the mean a particular value is.
- The z-score corresponding to 4% is -1.75.
- The z-score corresponding to 97% is 1.88.
2. Use the z-score formula to find the values of x at each z-score.
- For z = -1.75, x = (z * standard deviation) + mean.
- For z = 1.88, x = (z * standard deviation) + mean.
3. Form two equations using the given values and solve them to find the mean (μ) and standard deviation (σ).
ii. Finding the interquartile range:
1. Convert the given percentage values to z-scores using the standard normal distribution table or a calculator.
- The z-score corresponding to 25% is approximately -0.67.
- The z-score corresponding to 75% is approximately 0.67.
2. Use the z-score formula to find the values of x at each z-score.
- For z = -0.67, x = (z * standard deviation) + mean.
- For z = 0.67, x = (z * standard deviation) + mean.
3. Calculate the difference between the values obtained in step 2 to find the interquartile range.
Now let's calculate the mean, standard deviation, and interquartile range using the steps outlined above.