Question

Suppose that the mean cranial capacity for men is 1180 cc (cubic centimeters) and that the standard deviation is 300 cc. Assuming that men's cranial capacities are normally distributed, complete the following statements. (a) Approximately ___ of men have cranial capacities between 880 cc and 1480 cc(b) Approximately 95% of men have cranial capacities between ___ cc and ___ cc.

Answer 1

To calculate the probabilities of cranial capacities, we need to find the z-scores and use the standard normal distribution table. (a) The probability of men having cranial capacities between 880 cc and 1480 cc can be found by calculating the corresponding z-scores and finding the difference in probabilities. (b) Approximately 95% of men have cranial capacities between 580 cc and 1880 cc by calculating the lower and upper bounds using the formula: mean ± (2 * standard deviation).

Step-by-step explanation:

To find the probabilities, we need to calculate the z-scores for the given cranial capacities.

(a) The z-scores for 880 cc and 1480 cc can be calculated using the formula:

z = (x - μ) / σ, where x is the value, μ is the mean, and σ is the standard deviation.

Calculating the z-scores for 880 cc and 1480 cc, we get: z1 = (880 - 1180) / 300 = -1, z2 = (1480 - 1180) / 300 = 1.

Using the standard normal distribution table, we can find the corresponding probabilities for these z-scores. The probability of men having cranial capacities between 880 cc and 1480 cc is the difference between these probabilities.

(b) Approximately 95% of men have cranial capacities within 2 standard deviations of the mean. We can calculate the values using the formula: lower bound = mean - (2 * standard deviation), upper bound = mean + (2 * standard deviation). Substituting the given values, we get lower bound = 1180 - (2 * 300) = 580 cc, upper bound = 1180 + (2 * 300) = 1880 cc.

Answer 2

Step 1

Given;

`\begin{gathered} mean=\mu=1180 \\ Standard\text{ Deviation=}\sigma=300 \end{gathered}`

Step 2

Using the empirical rule;

[tex]P(\mu-\sigmaA) Hence, approximately 68% of men have cranial capacities between 880cc and 1480cc.[tex]\begin{gathered} P(\mu-2\sigmaB) Hence approximately 95% of men have cranial capacities between 580cc and 1780cc.