Question

The height of the female population is normally distributed with mean (p) cm and standard deviation (0) cm. It is known that 5% of the women are shorter than 154cm and 30% are taller than 172cm. (a) Compute for u and o in cm (b) A female is chosen at random from the population, find the probability that she is taller than 160cm.

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Answer 1

(a) The mean (μ) is 163.84 cm and the standard deviation (σ) is 7.52 cm.

(b) The probability that a randomly chosen female from the population is taller than 160 cm is approximately 0.6554 or 65.54%.

Step-by-step explanation:

The standard normal distribution is used to solve this problem. Firstly, to find the mean (μ) and standard deviation (σ), inverse normal distribution is applied to the given percentiles. Using the Z-score formula for the 5th percentile (Z_5 = -1.6449) and the 70th percentile (Z_30 = 0.5244), we can set up equations to find μ and σ.

For the 5th percentile: Z = (154 - μ) / σ, where Z = -1.6449.

For the 70th percentile: Z = (172 - μ) / σ, where Z = 0.5244.

Solving these equations simultaneously, we find μ = 163.84 cm and σ = 7.52 cm.

Next, to find the probability that a randomly chosen female is taller than 160 cm, we convert 160 cm to a Z-score using the formula Z = (X - μ) / σ. Substituting the values, Z = (160 - 163.84) / 7.52 ≈ -0.5126. Using a standard normal distribution table or calculator, we find the area to the right of Z = -0.5126, which gives a probability of approximately 0.6554 or 65.54%.

Therefore, the probability that a randomly chosen female from the population is taller than 160 cm is approximately 0.6554 or 65.54%.

Answer 2

To compute the values of mean (µ) and standard deviation (σ) for the height of the female population, use the z-score formula and the standard normal distribution table. To find the probability that a random female is taller than 160cm, calculate the z-score for this height and use the standard normal distribution table.

Step-by-step explanation:

To compute the values of mean (µ) and standard deviation (σ) for the height of the female population, we can use the z-score formula and the standard normal distribution table.

(a) For the given information, we have: 5% of women are shorter than 154cm, so the z-score corresponding to this probability is -1.645. Using the formula: z = (x - µ) / σ, we can write: -1.645 = (154 - µ) / σ. Similarly, 30% of women are taller than 172cm, so the z-score corresponding to this probability is 0.524. Using the formula again, we have: 0.524 = (172 - µ) / σ. Now, substitute μ back into one of the original equations to solve for σ:

163 cm - 1.645σ = 154 cm

-1.645σ = -9 cm

σ ≈ 5.47 cm

Therefore, the mean (μ) of the female height distribution is approximately 163 cm and the standard deviation (σ) is approximately 5.47 cm.

(b) To find the probability of a randomly chosen woman being taller than 160 cm, calculate the area to the right of 160 cm under the normal distribution with μ = 163 cm and σ = 5.47 cm.

Using a z-table or calculator, you can find this area to be approximately 0.6915.

Thus, the probability of a randomly chosen woman being taller than 160 cm is approximately 69.15%.