Hello!!
A survey indicates that for each trip to the supermarket, a shopper spends an average of 45 minutes with a standard deviation of 12 minutes in the store.the length is normally distributed and represented by variable x.
a.A find the probability that he will be in store for each interval of the time listed below
b.If 200 shoppers enter the store, how many shoppers will be in the store for each interval listed below:
between 24-54minutes
more than 39minutes
Hello! Let's dive into the supermarket fun, shall we?
a. To find the probability that the shopper will be in the store for each interval of time, we can use the z-score formula:
z = (x - μ) / σ
Where:
x is the value of interest
μ is the mean (average) time spent in the store (45 minutes)
σ is the standard deviation (12 minutes)
For each interval, we'll calculate the z-scores and then use a standard normal table to find the corresponding probabilities.
Interval 1: 24-54 minutes
To find the probability of spending between 24-54 minutes in the store, we need to find the z-scores for both endpoints:
z1 = (24 - 45) / 12
z2 = (54 - 45) / 12
Using a standard normal table or calculator, we find the corresponding probabilities:
P(24 < x < 54) = P(-1.75 < z < 0.75) ≈ 0.6725
Interval 2: More than 39 minutes
To find the probability of spending more than 39 minutes in the store, we calculate the z-score for the endpoint:
z = (39 - 45) / 12
Again, using a standard normal table or calculator, we find the corresponding probability:
P(x > 39) = P(z > -0.5) ≈ 0.6915
b. If 200 shoppers enter the store, we can use the probabilities we calculated in part a to estimate the number of shoppers in each interval.
For the interval 24-54 minutes:
Number of shoppers ≈ Total shoppers * Probability
Number of shoppers ≈ 200 * 0.6725 ≈ 134.5 (rounded to the nearest whole number)
For the interval more than 39 minutes:
Number of shoppers ≈ Total shoppers * Probability
Number of shoppers ≈ 200 * 0.6915 ≈ 138.3 (rounded to the nearest whole number)
So, approximately 134 shoppers will be in the store between 24-54 minutes, and around 138 shoppers will spend more than 39 minutes in the store.
Remember, these are just estimates based on probabilities, so the actual number may vary. Now go shop with a smile!
a. To find the probability, we need to use the normal distribution table or calculator. The formula for the z-score is:
z = (x - μ) / σ
where x is the value we are interested in, μ is the mean (45 minutes), and σ is the standard deviation (12 minutes).
1. Probability of being in the store between 24-54 minutes:
To find the probability of being in this interval, we need to calculate the z-scores for the lower and upper bounds of the interval.
For the lower bound (24 minutes):
z_lower = (24 - 45) / 12
z_lower = -1.75
For the upper bound (54 minutes):
z_upper = (54 - 45) / 12
z_upper = 0.75
Using the normal distribution table or calculator, we can find the probability for each z-score:
P(z <= -1.75) = 0.0401
P(z <= 0.75) = 0.7734
The probability of being in the store between 24-54 minutes is:
P(24 <= x <= 54) = P(z <= 0.75) - P(z <= -1.75)
P(24 <= x <= 54) = 0.7734 - 0.0401
P(24 <= x <= 54) = 0.7333
2. Probability of being in the store for more than 39 minutes:
To find this probability, we need to calculate the z-score for the upper bound (39 minutes):
z_upper = (39 - 45) / 12
z_upper = -0.5
Using the normal distribution table or calculator, we can find the probability:
P(z <= -0.5) = 0.3085
The probability of being in the store for more than 39 minutes is:
P(x > 39) = 1 - P(z <= -0.5)
P(x > 39) = 1 - 0.3085
P(x > 39) = 0.6915
b. If 200 shoppers enter the store, we can use the probabilities obtained above to estimate the number of shoppers falling into each interval.
1. Between 24-54 minutes:
Number of shoppers = P(24 <= x <= 54) * 200
Number of shoppers = 0.7333 * 200
Number of shoppers ≈ 146.67 (rounded to the nearest whole number)
So, approximately 147 shoppers will fall into this interval.
2. More than 39 minutes:
Number of shoppers = P(x > 39) * 200
Number of shoppers = 0.6915 * 200
Number of shoppers ≈ 138.3 (rounded to the nearest whole number)
So, approximately 138 shoppers will be in the store for more than 39 minutes.
Hello! I can help you with those questions. To solve them, we will use the concept of the normal distribution and the given information about the average time and standard deviation.
a. To find the probability that a shopper will be in the store for a specific interval of time, we need to calculate the area under the normal distribution curve for that interval.
1. Between 24-54 minutes:
To find the probability that the shopper will be in the store for 24-54 minutes, we need to calculate the cumulative probability from 24 to 54 minutes.
Z-score = (x - mean) / standard deviation
For 24 minutes:
Z1 = (24 - 45) / 12
For 54 minutes:
Z2 = (54 - 45) / 12
Next, we use a standard normal table or a calculator to find the cumulative probabilities associated with each Z-score. Then, we subtract the smaller cumulative probability from the larger one to find the probability in the desired interval.
P(24 < x < 54) = P(Z1 < Z < Z2)
b. If 200 shoppers enter the store, we can use the concept of the binomial distribution to determine how many shoppers will fall within each interval.
1. Between 24-54 minutes:
To find the number of shoppers that will be in the store for 24-54 minutes, we need to calculate the probability of each shopper falling within that interval and multiply it by the total number of shoppers (200).
P(24 < x < 54) * Total number of shoppers
2. More than 39 minutes:
To find the number of shoppers that will be in the store for more than 39 minutes, we need to calculate the probability of each shopper spending more than 39 minutes and multiply it by the total number of shoppers (200).
P(x > 39) * Total number of shoppers
By using the formulas mentioned above, we can calculate the probabilities and the number of shoppers for each interval.
a. I cannot respond adequately, since the times are not listed.
b. Z = (x-μ)/SD
For the last question, Z = (39-45)/12 = -.5
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion from that Z score that lies above that point. Multiply by 200.
Use a similar method to answer the remainder of your questions.
I hope this helps.