A box of tickets averages out to 75 and the SD is 10. One hundred draws are made at random with replacement from this box.
a)Find the chance that the average of the draws will be in the range 65 to 85.
b)repeat for the range 74 to 76.
Use z-scores and a z-table for this problem. Because you are given a sample size, you will need to include the sample size in the calculation:
z = (x - mean)/(sd/√n)
For a):
This will be very close to 100% chance that the average of the draws will be in the range listed (z-scores will be very large).
For b):
z = (74 - 75)/(10/√100)
z = (76 - 75)/(10/√100)
Finish the calculation for both z-scores. Use the z-table to find the probability between the two scores.
I hope this will help get you started.
To find the chance that the average of the draws falls within a given range, we can use the Central Limit Theorem (CLT). According to the CLT, the distribution of the sample means will approach a normal distribution as the sample size increases.
a) Range 65 to 85:
1. Calculate the standard error of the sample mean:
The standard error is the standard deviation of the population divided by the square root of the sample size.
Standard Error (SE) = Standard Deviation (SD) / √(sample size)
SE = 10 / √100 = 10 / 10 = 1
2. Convert the range values to z-scores:
To find the z-scores, we use the formula: z = (x - μ) / SE
For the lower range z-score:
z1 = (65 - 75) / 1 = -10
For the upper range z-score:
z2 = (85 - 75) / 1 = 10
3. Look up the probabilities associated with the z-scores:
Using a standard normal distribution table or a statistical calculator, find the probabilities for the z1 and z2 values.
P(z < -10) ≈ 0
P(z < 10) = 1
4. Calculate the final probability:
The probability of the average falls within the range 65 to 85 can be found by subtracting the probability from step 3.
P(65 < average < 85) = P(z < 10) - P(z < -10)
P(65 < average < 85) ≈ 1 - 0 = 1
b) Range 74 to 76:
1. Calculate the standard error as done in step 1 of the previous question:
SE = 10 / √100 = 1
2. Convert the range values to z-scores:
For the lower range z-score:
z1 = (74 - 75) / 1 = -1
For the upper range z-score:
z2 = (76 - 75) / 1 = 1
3. Look up the probabilities associated with the z-scores:
Using a standard normal distribution table or a statistical calculator, find the probabilities for the z1 and z2 values.
P(z < -1) ≈ 0.1587
P(z < 1) ≈ 0.8413
4. Calculate the final probability:
The probability of the average falls within the range 74 to 76 can be found by subtracting the probability from step 3.
P(74 < average < 76) = P(z < 1) - P(z < -1)
P(74 < average < 76) ≈ 0.8413 - 0.1587 = 0.6826
Therefore,
a) The chance that the average of the draws will be in the range 65 to 85 is approximately 100% (or 1).
b) The chance that the average of the draws will be in the range 74 to 76 is approximately 68.26%.