The tensile strength of paper used to make grocery bags is an important quality characteristic. It is known that the strength is normally distributed with mean 40 lb/in2 and standard deviation 2 lb/in2. The purchaser of the bags requires them to have a strength of at least 35 lb/in2.
a) Calculate the probability that bags produced from this paper will meet or exceed the specification. Hint: Use Normal_probabilities Excel template.
Answer
0062
Enter your answer to 4 decimal places.
b) Choose the correct notation for the probability being asked in a).
Answer
P(X≥35)
c) What is the tensile strength value that only 1 in 100 bags would exceed?
Answer
Enter your answer to 2 decimal places.
d) Use the dropdown menus below to describe the probability distribution of the average strength of paper from a sample of 25 of these bags.
Probability distribution of the average strength is Answer
Student’s T
, with parameters:Answer
n
= Answer
(enter your answer to 2 decimal places) and Answer
df
= Answer
(enter your answer to 2 decimal places).
e) What is the probability that this average is less than 39.5?
Answer
Enter your answer to 4 decimal places.
a) The probability that bags produced from this paper will meet or exceed the specification can be calculated using the z-score formula:
z = (X - μ) / σ
where X is the minimum required strength of 35 lb/in2, μ is the mean strength of 40 lb/in2, and σ is the standard deviation of 2 lb/in2.
Plugging in the values:
z = (35 - 40) / 2 = -2.5
Using the Normal_probabilities Excel template, the corresponding probability is 0.0062.
b) The correct notation for the probability being asked in a) is P(X ≥ 35).
c) To find the tensile strength value that only 1 in 100 bags would exceed, we need to find the corresponding z-score for the desired probability of 0.01 (since 1 - 0.01 = 0.99).
Using the z-score formula:
z = invNorm(0.99) = 2.326
Now, we can convert the z-score back to the actual strength value using:
X = μ + z * σ
X = 40 + 2.326 * 2 = 44.652
Therefore, the tensile strength value that only 1 in 100 bags would exceed is approximately 44.65 lb/in2.
d) The probability distribution of the average strength of paper from a sample of 25 bags can be modeled using the Student's T-distribution. The parameters for this distribution are the sample size (n = 25) and the degrees of freedom (df = n - 1 = 24).
Therefore, the probability distribution of the average strength is Student's T, with parameters:
n = 25
df = 24
e) To calculate the probability that this average is less than 39.5, we can use the Student's T-distribution with the given parameters.
Using the t-score formula:
t = (X - μ) / (σ / sqrt(n))
t = (39.5 - 40) / (2 / sqrt(25)) = -0.625
Using the t-distribution table or an online calculator, the corresponding probability is approximately 0.2676.
a) To calculate the probability that bags produced from this paper will meet or exceed the specification, we need to find the area under the normal distribution curve to the right of 35 lb/in^2. Using a normal distribution table or calculator, we can find this probability.
P(X ≥ 35) = 1 - P(X < 35)
Using the given mean (μ = 40 lb/in^2) and standard deviation (σ = 2 lb/in^2), we can compute the z-score for 35 lb/in^2:
z = (35 - 40) / 2 = -2.5
Now we can look up the z-score in a standard normal distribution table or use a calculator to find the corresponding probability.
P(Z < -2.5) ≈ 0.0062
Therefore, the probability that bags produced from this paper will meet or exceed the specification is approximately 0.0062.
b) The correct notation for the probability being asked in part a) is P(X ≥ 35), where X represents the tensile strength of the bags.
c) To find the tensile strength value that only 1 in 100 bags would exceed, we need to find the z-score that corresponds to a cumulative probability of 0.99.
Using the standard normal distribution table or calculator, we can find the z-score such that P(Z < z) = 0.99:
z ≈ 2.33
Now we can solve for the tensile strength value using the z-score formula:
35 + (z * 2) = 35 + (2.33 * 2) ≈ 39.66
Therefore, the tensile strength value that only 1 in 100 bags would exceed is approximately 39.66 lb/in^2.
d) The probability distribution of the average strength of paper from a sample of 25 bags is a Student's t-distribution with parameters n = 25 (sample size) and df = n - 1 = 25 - 1 = 24.
e) To find the probability that the average strength is less than 39.5, we need to find the area under the t-distribution curve to the left of 39.5.
Using the t-distribution table or calculator, we can calculate this probability. However, the given information does not provide the necessary parameters to determine the exact probability.
a) To calculate the probability that bags produced from this paper will meet or exceed the specification (have a strength of at least 35 lb/in2), we can use the standard normal distribution.
First, let's calculate the z-score:
z = (X - μ) / σ
where X is the specified strength (35 lb/in2), μ is the mean (40 lb/in2), and σ is the standard deviation (2 lb/in2).
z = (35 - 40) / 2 = -2.5
Next, we look up the probability corresponding to this z-score in a standard normal distribution table or use a statistical software/tool.
The probability that bags produced from this paper will meet or exceed the specification is P(Z ≥ -2.5).
b) The correct notation for the probability in part a) is P(X≥35), where X represents the tensile strength of the paper.
c) To find the tensile strength value that only 1 in 100 bags would exceed, we need to find the z-score corresponding to the desired probability.
From a standard normal distribution table or using statistical software, we find that the z-score corresponding to a cumulative probability of 0.99 (1 - 0.01) is approximately 2.33.
Next, we use the z-score formula to solve for the tensile strength:
z = (X - μ) / σ
Rearranging the formula, we get:
X = (z * σ) + μ
Plugging in the values:
X = (2.33 * 2) + 40 = 44.66
Therefore, the tensile strength value that only 1 in 100 bags would exceed is approximately 44.66 lb/in2.
d) The probability distribution of the average strength of paper from a sample of 25 bags follows the Student's T distribution. The parameters are n = 25 (sample size) and df (degrees of freedom).
e) To find the probability that the average strength is less than 39.5 lb/in2, we can use the Student's T distribution with the given parameters.
Using a statistical software/tool or lookup tables, we can calculate the probability P(T < 39.5) by inputting the appropriate degrees of freedom (df) and t-score (calculated using the sample mean, sample standard deviation, and sample size).
Please note that we don't have enough information to provide the exact values for parts d) and e).