Question

Fish of Lake Laengelmaevesi. An article by J. Puranen of the Department of Statistics, University of Helsinki, discussed a classic study on several variables of seven different species of fish caught in Lake Laengelmaevesi, Finland. On the Weiss Stats site, we present the data on weight (in grams) and length (in centimeters) from the nose to the beginning of the tail for four of the seven species. Perform the required parts for both the weight and length data.

a. Obtain individual normal probability plots and the standard deviation of the samples.

b. Perform a residual analysis.

c. Use your results from parts (a) and (b) to decide whether conducting a one-way ANOVA test on the data is reasonable. If so. also do parts (d) and (e).

d. Use a one-way ANOVA test to decide, at the $5\%$significance level, Whether the data provide sufficient evidence to conclude that a difference exists among the means of the populations fewer than the samples were taken.

e. Interpret your results from part (d)

Short answer

(a) The standard deviations are length Lahna, length Siika, length saerki, length Parkki

Length lahna $=3.594$, Length Siika $=5.58$

Length saerki $=3.46$, Length parkki $=3.28$

Weight lahna $206.6$, Weight Siika $=309.6$

Weight saerki $=88.8$, Weight parkki$=77.8$

(b)

(c) The assumptions of normal populations and equal population standard deviations are not met. Hence, conducting of ANOVA test on given data of length is reasonable.

(d) The ANOVA test is performed on given data.

(e) The data give enough information to establish that the mean lengths of the four fish species captured in the lake differ.

Step-by-step solution

Part (a) Step 1: Concept introduction

ANOVA, or non - parametric tests, is a potent quantitative approach for revealing the disparity amongst two or more phenomena or aspects using variable importance. It also illustrates how to determine statistical significance among several demographics.

Part (a) Step 2: Explanation

To plot the normal probability plots and determine their standard deviations, use any software.

Enter the specified length data.

The normal probability plots for the lengths Lahna, Siika, Saerki, and Parkki are shown below.

Enter the given data of weight

The normal probability plot of Lahna, weight Siika, weight saerki, weight Parkki is given below

The standard deviations are length Lahna, length Siika, length saerki, length Parkki

Length lahna$=3.594$, Length Siika $=5.58$

Length saerki $=3.46$, Length parkki $=3.28$

Weight lahna $=206.6$, Weight Siika $=309.6$

Weight saerki $=88.8$, Weight parkki $=77.8$

Part (b) Step 1: Concept introduction

ANOVA, or non - parametric tests, is a potent quantitative approach for revealing the disparity amongst two or more phenomena or aspects using variable importance. It also illustrates how to determine statistical significance among several demographics.

Part (b) Step 2: Explanation

To perform residual analysis, you can use any software.

Fill in the length and species information.

Draw the residual probability distribution.

Draw the plot of residual verses fits.

Enter the data of weight and species.

Draw the normal probability of residual

Draw the plot of residual verses fits.

Part (c) Step 1: Concept introduction

ANOVA, or non - parametric tests, is a potent quantitative approach for revealing the disparity between two or more phenomena or aspects using variable importance. It also illustrates how to determine statistical significance among several demographics.

Part (c) Step 2: Explanation

From parts a and b

The ratio of largest to smallest standard deviations is

$=\frac{5.581}{3.29}$

$=1.7<2$

The ratio must not exceed two.

It shows that the equal standard deviation assumption is not broken.

As a result, the assumptions of normal populations and equal standard deviations in the population are met.

Regarding weight,

The greatest to smallest standard deviation ratio is

$=\frac{309.6}{78.76}$

$=3.93>2$

The proportion is higher than two.

It shows that the assumption of equal standard deviation has been broken.

Normal population assumptions and equal population standard deviations are not met. As a result, performing an ANOVA test on length data is feasible.

Part (d) Step 1: Concept introduction

ANOVA, or non - parametric tests, is a potent quantitative approach for revealing the disparity amongst two or more phenomena or aspects using variable importance. It also illustrates how to determine statistical significance among several demographics.

Part (d) Step 2: Explanation

The level of significance is $\alpha =0.05$

Consider the test hypothesis

Null hypothesis

$H_0$: The data provide is not sufficient evidence to conclude that the mean lengths of the four species of fish caught in lake are different.

Alternative hypothesis

$H_a$: The data provide is not sufficient evidence to conclude that the mean lengths of the four species of fish caught in lake are different.

Enter the given data,

Then we get, $F-$static $=44.98$, $p-$value $=0$.

Hence, ANOVA test is performed on given data.

Part (e) Step 1: Concept introduction

ANOVA, or non - parametric tests, is a potent quantitative approach for revealing the disparity amongst two or more phenomena or aspects using variable importance. It also illustrates how to determine statistical significance among several demographics.

Part (e) Step 2: Explanation

The level of significance is $\alpha =0.05$

From part d

The value of p is $0$

The rejection rule

$p-value\le \alpha$, the null hypothesis is rejected

$p-value(0.0)<\alpha (0.05)$

At a $5\%$significant threshold, the null hypothesis is thus rejected.

As a result, the findings are statistically significant at the$5\%$ level.

As a result, the data give enough information to establish that the mean lengths of the four fish species captured in the lake differ.