Researchers are studying two populations of…

Mathematics Questions
Short Answer
Step by Step Solution
Mathematics Questions

Researchers are studying two populations of sea turtles. In population D, 30% of the turtles have a shell length greater than 2 feet, while in population E, 20% have a shell length greater than 2 feet. From a random sample of 40 turtles from D, 15 had a shell length greater than 2 feet. From a random sample of 60 turtles from E, 11 had a shell length greater than 2 feet. Let pÃÇ_D represent the sample proportion for D, and pÃÇ_E represent the sample proportion for E.

Short Answer

The sample proportions for turtles with shell lengths greater than 2 feet are 0.375 for population D and approximately 0.1833 for population E, resulting in a difference of 0.1917. The standard deviation of this difference is approximately 0.0914, leading to a z-score of about 1.0029, with a probability of the difference being greater than 0.1917 calculated as approximately 0.1580.

Step-by-Step Solution

Step 1: Calculate Sample Proportions

To find the sample proportions of turtles with shell lengths greater than 2 feet for both populations, use the formulas:

  • The sample proportion for population D, denoted as pÀÜD, is calculated as: pÀÜD = 15/40 = 0.375.
  • The sample proportion for population E, denoted as pÀÜE, is calculated as: pÀÜE = 11/60 ‚âà 0.1833.
  • The difference between these proportions is (pÀÜD – pÀÜE) = 0.375 – 0.1833 = 0.1917.

Step 2: Determine Mean and Standard Deviation

The mean of the difference in sample proportions is given as 0.1917. To compute the standard deviation, apply the formula:

  • Standard deviation is calculated using: SD = ‚àö[(N_D * pÀÜD * (1 – pÀÜD) + N_E * pÀÜE * (1 – pÀÜE))].
  • Plugging in the values gives: SD = ‚àö[(40 * 0.375 * (1 – 0.375) + 60 * 0.1833 * (1 – 0.1833))] ‚âà 0.0914.

Step 3: Calculate the Probability

To find the probability that the difference in sample proportions is greater than 0.1917, calculate the z-score and the associated probability:

  • Use the formula Z = (pÀÜD – pÀÜE – (p_D – p_E)) / SD to find the z-score.
  • This leads to: Z = (0.1917 – 0.1) / 0.0914 ‚âà 1.0029.
  • Finally, find the probability: P(Z > 1.0029) = 1 – P(Z ‚⧠1.0029) ‚âà 0.1580.

Related Concepts

Sample proportion

The ratio of the number of successes to the total number of trials in a sample

Standard deviation

A measure of the amount of variation or dispersion of a set of values

Z-score

A statistical measurement that describes a value’s relation to the mean of a group of values, expressed in terms of standard deviations.

Table Of Contents
  1. Researchers are studying two populations of sea turtles. In population D, 30% of the turtles have a shell length greater than 2 feet, while in population E, 20% have a shell length greater than 2 feet. From a random sample of 40 turtles from D, 15 had a shell length greater than 2 feet. From a random sample of 60 turtles from E, 11 had a shell length greater than 2 feet. Let pÃÇ_D represent the sample proportion for D, and pÃÇ_E represent the sample proportion for E.

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