A grain mill manufactures 100-pound bags of flour for sale in restaurant supply warehouses. Historically, the weights of bags of flour manufactured at the mill were normally distributed with a mean of 100 pounds and a standard deviation of 15 pounds.
a. What is the probability that the weight of a randomly selected bag of flour falls between 94 and 106 pounds?
b. if samples of 36 bags are taken, what is the standard error of the mean?
c.what is the probability that a sample of 36 bags of flour has a mean weight between 94 and 106?
a. Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z scores.
b. SEm = SD/√n
c. Z = (score-mean)/SEm
Use same table.
a. To find the probability that the weight of a randomly selected bag falls between 94 and 106 pounds, we need to calculate the z-scores for these values and then find the corresponding probabilities using the standard normal distribution.
The z-score formula is given by: z = (x - μ) / σ
Where:
- x is the value we want to find the probability for (94 and 106)
- μ is the mean (100 pounds)
- σ is the standard deviation (15 pounds)
For 94 pounds:
z1 = (94 - 100) / 15 = -0.4
For 106 pounds:
z2 = (106 - 100) / 15 = 0.4
Now, we need to find the probability using the standard normal distribution table or a calculator. The probability of z1 and z2 corresponds to the area under the curve between these z-scores.
Using a z-table or calculator, we find that the probability of a z-score of -0.4 is approximately 0.3446, and the probability of a z-score of 0.4 is also 0.3446.
To find the probability between these two z-scores, we subtract the probability corresponding to the lower z-score from the probability corresponding to the higher z-score:
Probability = 0.3446 - 0.3446 = 0
Therefore, the probability that the weight of a randomly selected bag of flour falls between 94 and 106 pounds is approximately 0 (or 0%).
b. To calculate the standard error of the mean when samples of 36 bags are taken, we can use the formula:
SE = σ / √n
Where:
- SE is the standard error of the mean
- σ is the standard deviation (15 pounds)
- n is the sample size (36 bags)
Calculating the standard error:
SE = 15 / √36 = 15 / 6 = 2.5 pounds
Therefore, the standard error of the mean is 2.5 pounds.
c. To find the probability that a sample of 36 bags of flour has a mean weight between 94 and 106 pounds, we can use the Central Limit Theorem. According to the theorem, when the sample size is sufficiently large (usually greater than 30), the distribution of sample means will be approximately normal regardless of the shape of the original population distribution.
In this case, the sample mean is normally distributed with the same mean as the population mean (100 pounds) but with a smaller standard deviation, known as the standard error of the mean (2.5 pounds).
To find the probability, we can calculate the z-scores for the lower and upper bounds of the sample mean:
For 94 pounds:
z1 = (94 - 100) / 2.5 = -2.4
For 106 pounds:
z2 = (106 - 100) / 2.5 = 2.4
Using a z-table or calculator, we find that the probability of a z-score of -2.4 is approximately 0.0082, and the probability of a z-score of 2.4 is also 0.0082.
To find the probability between these two z-scores, we subtract the probability corresponding to the lower z-score from the probability corresponding to the higher z-score:
Probability = 0.0082 - 0.0082 = 0
Therefore, the probability that a sample of 36 bags of flour has a mean weight between 94 and 106 pounds is approximately 0 (or 0%).