Question

Researchers studying starfish collected two independent random samples of 40 starfish. One sample came from an ocean area in the north, and the other sample came from an ocean area in the south. Of the 40 starfish from the north, 6 were found to be over 8 inches in length. Of the 40 starfish from the south, 11 were found to be over 8 inches in length. Which of the following is the test statistic for the appropriate test to investigate whether there is a difference in proportion of starfish over 8 inches in length in the two ocean areas (north minus south)?A. 6â11640+1140âB. 6â110.1540+0.27540âC. 0.15â0.275(0.15)(0.275)(140+140)âD. 0.15â0.275(0.2125)(0.7875)(140+140)âE. 0.15â0.275(0.2125)(0.7875)140+140â

Answer 1

To determine the test statistic for investigating whether there is a significant difference in the proportion of starfish over 8 inches in length between the two ocean areas, we can use a two-proportion z-test. Here's how we calculate it step by step.

First, we calculate the proportion of starfish over 8 inches in length in each sample:
- For the north area: 6 out of 40 starfish are over 8 inches, so the proportion is \( \frac{6}{40} = 0.15 \).
- For the south area: 11 out of 40 starfish are over 8 inches, so the proportion is \( \frac{11}{40} = 0.275 \).

Next, we calculate the combined proportion of starfish over 8 inches in length for both areas together since this will be used in the standard error calculation:
- The combined sample size is \( 40 + 40 = 80 \).
- The combined number of starfish over 8 inches is \( 6 + 11 = 17 \).
- The combined proportion is \( \frac{17}{80} = 0.2125 \).

Now we calculate the standard error of the difference in proportions using the combined proportion:
- The formula for the standard error of the difference between two proportions is \(\sqrt{p_c (1 - p_c) \left(\frac{1}{n_1} + \frac{1}{n_2}\right)}\), where \(p_c\) is the combined proportion and \(n_1\) and \(n_2\) are the sample sizes.
- Substituting the values we have: \( \sqrt{0.2125 \cdot (1 - 0.2125) \left(\frac{1}{40} + \frac{1}{40}\right)} \).

Finally, we calculate the test statistic:
- The test statistic for the two-proportion z-test is \(\frac{p_1 - p_2}{SE}\), where \(p_1\) and \(p_2\) are the sample proportions and \(SE\) is the standard error calculated above.
- Subtracting the proportions: \( 0.15 - 0.275 = -0.125 \).
- The standard error has been calculated based on the combined proportion.
- Dividing the difference by the standard error, we find the value of the test statistic is approximately \(-1.3665334361513133\).

Thus, the test statistic for the hypothesis test investigating the difference in proportion of starfish over 8 inches between the two ocean areas is approximately -1.37.

The correct answer matches option C: \( 0.15 - 0.275\sqrt{(0.15)(0.275)\left(\frac{1}{40}+\frac{1}{40}\right)} \). This option matches the form of the calculation we did, but the actual numerical value given for the test statistic will guide us to select it as the best representation of our calculation.

Answer 2

The test statistic for the appropriate test is
`\frac{0.15-0.275}{\sqrt{0.2125(1-0.2125)[(1)/(40)+(1)/(40)]}}`.

Explanation:

The experiment conducted here is to determine whether there is a difference in proportion of starfish over 8 inches in length in the two ocean areas, i.e. north and south.

The hypothesis to test this can be defined as follows:

H₀: There is no difference in proportion of starfish over 8 inches in length in the two ocean areas, i.e. p₁ = p₂.

Hₐ: There is a difference in proportion of starfish over 8 inches in length in the two ocean areas, i.e. p₁ ≠ p₂.

The two-proportion z-test would be used to perform the test.

A sample of n = 40 starfishes are selected from both the ocean areas.

It provided that of the 40 starfish from the north, 6 were found to be over 8 inches in length and of the 40 starfish from the south, 11 were found to be over 8 inches in length.

Compute the sample proportion of starfish from north that were over 8 inches in length as follows:

`\hat p_(n)=(6)/(40)=0.15`

Compute the sample proportion of starfish from south that were over 8 inches in length as follows:

`\hat p_(s)=(11)/(40)=0.275`

The test statistic is:

`z=\frac{\hat p_(n)-\hat p_(s)}{\sqrt{P(1-P)[(1)/(n_(n))+(1)/(n_(s))]}}`

Compute the combined proportion P as follows:

`P=(X_(n)+X_(s))/(n_(n)+n_(s))=(6+11)/(40+40)=0.2125`

Compute the test statistic value as follows:

`z=\frac{\hat p_(n)-\hat p_(s)}{\sqrt{P(1-P)[(1)/(n_(n))+(1)/(n_(s))]}}`

`=\frac{0.15-0.275}{\sqrt{0.2125(1-0.2125)[(1)/(40)+(1)/(40)]}}`

Thus, the test statistic for the appropriate test is
`\frac{0.15-0.275}{\sqrt{0.2125(1-0.2125)[(1)/(40)+(1)/(40)]}}`.