Question

Assume that the traffic to the web site of Smiley’s People, Inc., which sells customized T-shirts, follows a normal distribution, with a mean of 4.42 million visitors per day and a standard deviation of 800,000 visitors per day.

Assume that the traffic to the web site of Smiley’s People, Inc., which sells customized T-shirts, follows a normal distribution, with a mean of 4.42 million visitors per day and a standard deviation of 800,000 visitors per day.
(a) What is the probability that the web site has fewer than 5 million visitors in a single day? If needed, round your answer to four decimal digits.
(b) What is the probability that the web site has 3 million or more visitors in a single day? If needed, round your answer to four decimal digits.
(c) What is the probability that the web site has between 3 million and 4 million visitors in a single day? If needed, round your answer to four decimal digits.
(d) Assume that 85% of the time, the Smiley’s People web servers can handle the daily web traffic volume without purchasing additional server capacity. What is the amount of web traffic that will require Smiley’s People to purchase additional server capacity? If needed, round your answer to two decimal digits.

Answer 1

To solve these probability questions, we will use the given mean, standard deviation, and the properties of the normal distribution. The z-score formula will help us standardize the values.

(a) To find the probability that the web site has fewer than 5 million visitors in a single day, we need to find the area under the normal curve to the left of 5 million.

z = (x - μ) / σ
z = (5,000,000 - 4,420,000) / 800,000

Using a z-score table or calculator, we find that the z-score is approximately 0.7257.

Now, we can find the probability by looking up the z-score in the standard normal distribution table (cumulative distribution function, CDF), or using a calculator:

P(X < 5,000,000) = P(Z < 0.7257) ≈ 0.7673

Therefore, the probability that the web site has fewer than 5 million visitors in a single day is approximately 0.7673.

(b) To find the probability that the web site has 3 million or more visitors in a single day, we need to find the area under the normal curve to the right of 3 million.

z = (x - μ) / σ
z = (3,000,000 - 4,420,000) / 800,000

Using the z-score table or calculator, we find that the z-score is approximately -1.775.

P(X ≥ 3,000,000) = P(Z ≥ -1.775)

Since the total area under the curve is 1, we can subtract the area to the left of -1.775 from 1:

P(X ≥ 3,000,000) = 1 - P(Z < -1.775)

Using the z-score table or calculator, we find P(Z < -1.775) ≈ 0.0384.

P(X ≥ 3,000,000) = 1 - 0.0384 ≈ 0.9616

Therefore, the probability that the web site has 3 million or more visitors in a single day is approximately 0.9616.

(c) To find the probability that the web site has between 3 million and 4 million visitors in a single day, we need to find the area under the normal curve between these two values.

We already calculated the z-score for 3 million visitors in part (b). Now, we'll calculate the z-score for 4 million visitors:

z1 = (3,000,000 - 4,420,000) / 800,000
z1 ≈ -1.775

z2 = (4,000,000 - 4,420,000) / 800,000
z2 ≈ -0.525

Using the z-score table or calculator, we find P(Z < -1.775) ≈ 0.0384 and P(Z < -0.525) ≈ 0.2995.

P(3,000,000 ≤ X ≤ 4,000,000) = P(-1.775 ≤ Z ≤ -0.525)
≈ P(Z < -0.525) - P(Z < -1.775)
≈ 0.2995 - 0.0384

Therefore, the probability that the web site has between 3 million and 4 million visitors in a single day is approximately 0.2611.

(d) To find the amount of web traffic that will require Smiley’s People to purchase additional server capacity, we need to find the value (x) such that 85%