In this topic you approximate probabilities for totals, counts, and averages of many independent pieces. You compute the mean and standard deviation of the sum, standardize, and look up the result in the normal table. This replaces exact calculations that are too long to do by hand. A problem belongs here when it asks about a total, count, or average of many independent pieces, and the number of pieces is large. You will practice: Standardizing a sum, Continuity correction, Normal approximation to the Poisson, Reading the table backwards.
In a pea-plant cross, each offspring is tall with probability 1/2, independently of the others. A greenhouse raises 100 offspring. Using the normal approximation with continuity correction, approximate the probability that at least 55 of them are tall.
Answer:
The count of tall plants is binomial with n = 100 and p = 1/2.
Apply the continuity correction for at least 55 tall plants.
A file server must process a batch of 400 independent requests. Each request takes a random processing time with mean 2 ms and standard deviation 0.5 ms. Use the central limit theorem to approximate the probability that the total processing time exceeds 812 ms.
Answer:
The total time is approximately normal by the CLT.
No continuity correction is needed since processing times are continuous.
A construction crane lifts a pallet holding 64 bags of cement. Each bag's weight is independent with mean 25 kg and standard deviation 4 kg. Use the central limit theorem to approximate the probability that the total load exceeds 1632 kg. You may use Φ(1) = 0.8413.
Answer:
The total weight has mean 1600 kg and standard deviation 32 kg because variances add.
Standardize the threshold of 1632 kg.
A clinic vaccinates 225 people. Each recipient independently reports a mild side effect with probability 0.2. Using the normal approximation with continuity correction, estimate the probability that at least 53 people report a side effect. You may use Φ(1.25) = 0.8944.
Answer:
The count of side effects is binomial with n = 225 and p = 0.2.
Apply the continuity correction for at least 53 reports.
A campus coffee cart serves 100 customers in a morning. Each customer's spend is independent with mean 6 € and standard deviation 2 €. Use the central limit theorem to approximate the probability that the morning's total revenue exceeds 620 €. (Use Φ(1) = 0.8413.)
Answer:
The total spend has mean 600 and standard deviation 20.
Standardize the threshold of 620.
The number of calls reaching a telephone helpline in one hour is Poisson distributed with mean 100. Using the normal approximation with continuity correction, estimate the probability that at most 110 calls arrive in a given hour. (Use Φ(1.05) = 0.8531.)
Answer:
A Poisson count with mean 100 is approximately normal with standard deviation 10.
Apply the continuity correction for at most 110 calls.
An orchard worker packs a crate with 25 apples. Apple weights are independent, each with mean 160 g and standard deviation 20 g. Let S be the total weight of the crate in grams. Find the standard deviation of S.
Answer:
Variances of independent weights add, so the SD of the sum is √n times the SD of one apple.
A bookshop makes 64 sales in one day. Each sale amount is independent with mean 15 € and standard deviation 4 €. The day's total revenue R therefore has mean 960 € and standard deviation 32 €. Compute the z-score of the value R = 1024 €.
Answer:
Standardize by subtracting the mean and dividing by the standard deviation.
A tutor grades 25 exam essays. Grading times are independent, each with mean 10 minutes and standard deviation 3 minutes. Use the central limit theorem to approximate the probability that the total grading time is at most 265 minutes. Use Φ(1) ≈ 0.8413.
Answer:
The total T has mean 250 and SD 15. Standardize and read the table.
A gardener plants 150 wildflower seeds. Each seed germinates independently with probability 0.4. Let X be the number of seeds that germinate. Using the normal approximation with continuity correction, approximate P(X ≤ 67). Use Φ(1.25) ≈ 0.8944.
Answer:
X is binomial with mean 60 and SD 6. Apply the continuity correction at 67.5.
During storm season, a weather station counts lightning strikes in its region. The number N of strikes in one week is Poisson distributed with mean 49. Using the normal approximation with continuity correction, approximate P(N ≤ 59). Use Φ(1.5) ≈ 0.9332.
Answer:
A Poisson with mean 49 has SD 7. Approximate by a normal and correct at 59.5.
A removal company loads 100 moving boxes into a truck. Box weights are independent, each with mean 20 kg and standard deviation 5 kg. Let S be the total weight in kg. Using the central limit theorem, find the value c such that P(S ≤ c) ≈ 0.975. Use Φ(1.96) = 0.975.
Answer:
S has mean 2000 and SD 50. Read the table backwards at probability 0.975, which gives z = 1.96.
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