I need help on this question i am confused. I don't think the answers i got are right
The library checks out an average of 320 books per day, with a standard deviation of 75 books. Suppose a simple random sample of 30 days is selected for observation.
What is the probability that the sample mean for the 30 days will be between 300 and 340 books?
0.4279
0.2128
0.8558
0.1064
0.9279
There is a 90% chance that the sample mean will fall below how many books?
Answer
323.5
416.0
342.5
337.5
443.4
To solve this question, we need to use the concept of the sample mean and the sampling distribution. The sample mean is calculated by taking the average of the observations in a sample, and the sampling distribution represents the distribution of all possible sample means.
Step 1: Calculate the standard error of the sample mean.
The standard error (SE) of the sample mean is calculated by dividing the standard deviation (σ) of the population by the square root of the sample size (n).
SE = σ / √n
Given that the standard deviation of the population is 75 books and the sample size is 30 days, we can calculate the standard error:
SE = 75 / √30
SE ≈ 13.691
Step 2: Convert the given values into z-scores.
To work with the standard normal distribution, we need to convert the given sample mean values (300 and 340) into z-scores. The z-score represents the number of standard deviations an observation is away from the mean.
z1 = (300 - 320) / SE
z1 ≈ -1.460
z2 = (340 - 320) / SE
z2 ≈ 1.460
Step 3: Find the probability between the z-scores.
Now that we have the z-scores, we can use a standard normal distribution table or a calculator to find the probability between these two z-scores.
Using a standard normal distribution table or calculator, find the probability associated with z = -1.460 and z = 1.460.
The probability that the sample mean falls between 300 and 340 books is approximately 0.8558.
Therefore, the correct answer is 0.8558 from the given options.
For the second part of the question:
Step 4: Find the z-score for the desired probability.
In this case, we need to find the z-score that corresponds to a 90% chance of the sample mean falling below a certain number of books. We can use a standard normal distribution table or calculator to find this z-score.
The z-score for a 90% chance (1 - 0.90 = 0.10) is approximately 1.282.
Step 5: Convert the z-score back into the number of books.
To convert the z-score back into the number of books, we can use the formula:
X = μ + (z * SE)
Given that the mean (μ) is 320 books and the standard error (SE) is 13.691, we can calculate the number of books:
X = 320 + (1.282 * 13.691)
X ≈ 337.480
Therefore, there is a 90% chance that the sample mean will fall below approximately 337.5 books.
The correct answer is 337.5 from the given options.