Researchers are studying two populations of sea turtles. In population …

Mathematics Questions
Short Answer
Step by Step Solution
Mathematics Questions

Researchers are studying two populations of sea turtles. In population D, 30 percent of the turtles have a shell length greater than 2 feet. In population E, 20 percent of the turtles have a shell length greater than 2 feet. From a random sample of 40 turtles selected from D, 15 had a shell length greater than 2 feet. From a random sample of 60 turtles selected from E, 11 had a shell length greater than 2 feet. Let pˆD represent the sample proportion for D, and let pˆE represent the sample proportion for E.

Short Answer

The difference in sample proportions between populations D and E is 0.1917, with a mean of 0.1917 and a standard deviation of 0.0914. Additionally, both conditions for normality of the sampling distribution are satisfied, confirming its approximate normality.

Step-by-Step Solution

a) Calculate the Difference in Sample Proportions

To find the difference between the sample proportions, we first need to calculate each proportion. For population D, the sample proportion p√a¬UD is obtained by dividing the number of turtles with a shell length greater than 2 feet (15) by the total sample size (40), yielding a result of 0.375. For population E, the sample proportion p√a¬UE is the number of turtles with a shell length greater than 2 feet (11) divided by the total sample size (60), resulting in 0.1833. The difference is then calculated as follows:

  • Difference: p‚àöa¬¨UD – p‚àöa¬¨UE = 0.375 – 0.1833 = 0.1917

b) Mean and Standard Deviation of the Sampling Distribution

The mean of the sampling distribution for the difference in sample proportions is directly obtained from the previously calculated difference, which is 0.1917. To compute the standard deviation, we use the formula that combines the proportions and their sample sizes. This calculation gives us a standard deviation of 0.0914. Thus, the values are:

  • Mean: 0.1917
  • Standard Deviation: 0.0914

c) Check Normality of the Sampling Distribution

To determine if the sampling distribution of the difference in sample proportions is approximately normal, we check if both np and n(1-p) conditions are satisfied for each population. Specifically, we verify if np ‚a• 10 and n(1-p) ‚a• 10 for both p√a¬UD and p√a¬UE. Since our sample sizes and calculated proportions meet this criterion, we confirm that the distribution is approximately normal.

  • Both conditions are satisfied: Yes

Related Concepts

Sample Proportion

The ratio of the number of successes (e.g., turtles with shell length greater than 2 feet) to the total number of observations in a sample, often denoted as pac.

Sampling Distribution

A probability distribution of all possible sample statistics (like sample means or proportions) that can be obtained from a population, which allows for inference about the population from sample data.

Normality Condition

A condition in statistics that checks if the distribution of a sampling statistic (such as means or proportions) approximates a normal distribution, typically verified by ensuring that np and n(1-p) are both greater than or equal to 10.

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

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