Questions: Which of these variables is most likely to have a Normal distribution? Number of heart attacks experienced by men before the age of 65 Lengths of newly hatched pythons Test scores on a very easy test Weight category (underweight, normal weight, overweight, obese) of adults

Which of these variables is most likely to have a Normal distribution?
Number of heart attacks experienced by men before the age of 65
Lengths of newly hatched pythons
Test scores on a very easy test
Weight category (underweight, normal weight, overweight, obese) of adults

Transcript text: Which of these variables is most likely to have a Normal distribution? Number of heart attacks experienced by men before the age of 65 Lengths of newly hatched pythons Test scores on a very easy test Weight category (underweight, normal weight, overweight, obese) of adults

Solution

Solution Steps

Step 1: Identify the Variable

Among the given variables, the one most likely to have a Normal distribution is the lengths of newly hatched pythons. This is because biological measurements often follow a Normal distribution due to the Central Limit Theorem.

Step 2: Define Parameters

For the analysis, we assume the following parameters for the lengths of newly hatched pythons:

Mean (\( \mu \)): 50 cm
Standard Deviation (\( \sigma \)): 5 cm
Sample Size (\( n \)): 30
Range: 45 cm to 55 cm

Step 3: Calculate Z-scores

The Z-scores for the specified range are calculated as follows:

For the lower bound (45 cm): \[ Z_{start} = \frac{45 - 50}{5} = -1.0 \]
For the upper bound (55 cm): \[ Z_{end} = \frac{55 - 50}{5} = 1.0 \]

Step 4: Calculate Probability

The probability that the sample mean falls within the specified range is given by: \[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(1.0) - \Phi(-1.0) = 0.6827 \] This indicates that there is a 68.27% chance that the sample mean of the lengths of newly hatched pythons will fall between 45 cm and 55 cm.