Assume that women’s heights are normally distributed with a mean given by μ = 63.4 in, and a standard deviation given by σ = 2.3 in. (a) If 1 woman is randomly selected, find the probability that her height is less than 64 in. (b) If 43 women are randomly selected, find the probability that they have a mean height less than 64 in.
a. z = (64 - 63.4)/2.3
b. z = (64 - 63.4)/(2.3/Sqrt(43))
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(a) To find the probability that a randomly selected woman has a height less than 64 inches, we need to calculate the area under the normal curve to the left of this value.
Step 1: Standardize the value 64 using the formula: z = (x - μ) / σ
where x is the value (64 inches), μ is the mean (63.4 inches), and σ is the standard deviation (2.3 inches).
z = (64 - 63.4) / 2.3
z = 0.26
Step 2: Look up the standardized value of 0.26 in the z-table or use a calculator. The cumulative probability is typically provided, so we need to find P(Z < 0.26).
Using the z-table or calculator, we find that P(Z < 0.26) ≈ 0.603
Therefore, the probability that a randomly selected woman has a height less than 64 inches is approximately 0.603 or 60.3%.
(b) To find the probability that a sample of 43 women has a mean height less than 64 inches, we need to calculate the standard error of the mean (SEM) and then use the Central Limit Theorem.
Step 1: Calculate the standard error of the mean using the formula: SEM = σ / √n
where σ is the standard deviation (2.3 inches) and n is the sample size (43).
SEM = 2.3 / √43
SEM ≈ 0.350
Step 2: Standardize the value 64 using the formula: z = (x̄ - μ) / SEM
where x̄ is the sample mean, μ is the population mean, and SEM is the standard error of the mean.
z = (64 - 63.4) / 0.350
z ≈ 1.71
Step 3: Look up the standardized value of 1.71 in the z-table or use a calculator. The cumulative probability is typically provided, so we need to find P(Z < 1.71).
Using the z-table or calculator, we find that P(Z < 1.71) ≈ 0.955.
Therefore, the probability that a randomly selected sample of 43 women has a mean height less than 64 inches is approximately 0.955 or 95.5%.
To solve both parts of this question, we will need to use the properties of the normal distribution and the Z-score.
(a) To find the probability that a randomly selected woman's height is less than 64 inches, we can use the Z-score formula:
Z = (X - μ) / σ
where:
Z = the standard score
X = the height value
μ = the mean
σ = the standard deviation
We want to find the probability of X < 64. We can calculate the Z-score as follows:
Z = (64 - 63.4) / 2.3 = 0.26
Now, we can use a standard normal distribution table or a calculator to find the probability associated with a Z-score of 0.26. The probability is equivalent to the area under the normal curve to the left of the Z-score.
Using a standard normal table or calculator, the probability corresponding to a Z-score of 0.26 is approximately 0.6026.
Therefore, the probability that a randomly selected woman's height is less than 64 inches is approximately 0.6026 or 60.26%.
(b) To find the probability that 43 randomly selected women have a mean height less than 64 inches, we need to use the Central Limit Theorem, which states that the distribution of the sample means will be approximately normally distributed, regardless of the shape of the population distribution, as long as the sample size is large enough.
For this calculation, we need to find the Z-score for the sample mean using the formula:
Z = (X̄ - μ) / (σ / √n)
where:
Z = the standard score for the sample mean
X̄ = the sample mean
μ = the population mean
σ = the population standard deviation
n = the sample size
In this case, X̄ = 64, μ = 63.4, σ = 2.3, and n = 43.
Z = (64 - 63.4) / (2.3 / √43) = 0.97
Now, we can use a standard normal table or calculator to find the probability associated with a Z-score of 0.97. As before, the probability is equivalent to the area under the normal curve to the left of the Z-score.
Using a standard normal table or calculator, the probability corresponding to a Z-score of 0.97 is approximately 0.832.
Therefore, the probability that 43 randomly selected women have a mean height less than 64 inches is approximately 0.832 or 83.2%.