5) The product is sold in packets whose masses are normally distributes with mean of 1.42 kg and a standard deviation of 0.025 kg.
(1) Find the probability that the mass of a packet, selected at random, lies between 1.37 kg and 1.45 kg
To find the probability that the mass of a packet lies between 1.37 kg and 1.45 kg, we need to calculate the z-scores corresponding to these two values and then use the z-table to find the probability.
Step 1: Calculate the z-scores
The z-score formula is given by: z = (x - μ) / σ
where x is the value, μ is the mean, and σ is the standard deviation.
For 1.37 kg:
z1 = (1.37 - 1.42) / 0.025
For 1.45 kg:
z2 = (1.45 - 1.42) / 0.025
Step 2: Look up the z-scores in the z-table
The z-table provides the cumulative probability up to a given z-score. We need to find the cumulative probabilities for z1 and z2.
Using the z-table, let's assume that z1 corresponds to a cumulative probability of P1 and z2 corresponds to P2.
Step 3: Calculate the probability between the two z-scores
To find the probability between the two z-scores, we subtract the cumulative probability associated with z1 from the cumulative probability associated with z2.
P1 = cumulative probability corresponding to z1
P2 = cumulative probability corresponding to z2
Required probability = P2 - P1
Let's calculate the z-scores and find P1 and P2.
For 1.37 kg:
z1 = (1.37 - 1.42) / 0.025 = -2
Looking up the z-table, the cumulative probability for z = -2 is approximately 0.0228.
So, P1 = 0.0228.
For 1.45 kg:
z2 = (1.45 - 1.42) / 0.025 = 1.2
Looking up the z-table, the cumulative probability for z = 1.2 is approximately 0.8849.
So, P2 = 0.8849.
Required probability = P2 - P1
= 0.8849 - 0.0228
= 0.8621
Therefore, the probability that the mass of a packet selected at random lies between 1.37 kg and 1.45 kg is approximately 0.8621.
To find the probability that the mass of a packet lies between 1.37 kg and 1.45 kg, we need to find the area under the normal distribution curve between these two values.
Here's the step-by-step process to calculate it:
Step 1: Standardize the values.
To use the standard normal distribution table, we need to convert the given values into z-scores. The formula to calculate the z-score is:
z = (x - μ) / σ
Where:
x = the given value
μ = the mean of the distribution
σ = the standard deviation of the distribution
So, for 1.37 kg:
z1 = (1.37 - 1.42) / 0.025
And for 1.45 kg:
z2 = (1.45 - 1.42) / 0.025
Step 2: Look up the z-scores in the standard normal distribution table.
You can use a standard normal distribution table or a calculator with a normal distribution function to find the probability associated with each z-score.
For example, using a table or calculator, you can find:
P(z1) = probability corresponding to z1
P(z2) = probability corresponding to z2
Step 3: Calculate the desired probability.
The probability between the two values is given by the difference of the two probabilities:
P(1.37 kg ≤ X ≤ 1.45 kg) = P(z1 ≤ Z ≤ z2) = P(z2) - P(z1)
Finally, substitute the values of P(z1) and P(z2) to calculate the probability that the mass of a packet lies between 1.37 kg and 1.45 kg.
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 between the two Z scores.