Question

Refer to Exercise 5.

(a) Interpret the value of $S{E}_b$ in context.

(b) Find the critical value for a 90% confidence interval for the slope of the true regression line. Then calculate the confidence interval. Show your work.

(c) Interpret the interval from part (b) in context.

(d) Explain the meaning of “90% confident” in context.

Short answer

(a) ${\mathrm{SE}}_b=0.002456$

(b) The confidence interval is $(0.004244,0.012894)$

(c) As a result, we have 90 percent confldence that the true slope of the population regression line is between $(0.004244,0.012894)$

(d) "confident" means that the confidence interval will contain the correct value of the population parameter in $90$ of all available samples.

Step-by-step solution

Part (a)Step 1: Given information

Given in the question that, We need to Interpret the value of $S{E}_b$in context .

Part(a) Step 2: Explanation

In order to answer the topic, researchers looked at how nature manages the size of animal predators that destroy a bigger proportion of the population when prey density is low. The amount of perch in a confined region is the explanatory variable, and the proportion of perch killed by bass in two hours while bass have access to the perch is the response variable. A scatterplot is provided, which depicts a linear relationship. Also included is a residual graphic for the least square regression line. For this data, the computer output is provided. The standard deviation of the slope is supplied in the row with "Perch" and the column with "Stdev.":${\mathrm{SE}}_b$

${\mathrm{SE}}_b=0.002456$

This indicates that the slope is steep.

Part(b) Step 1: Given information

We need to interpret the interval from part (b) in context.

Part (b) Step 2: Explanation

In order to answer the topic, researchers looked at how nature manages the size of animal predators that destroy a bigger proportion of the population when prey density is low. The amount of perch in a confined region is the explanatory variable, and the proportion of perch killed by bass in two hours while bass have access to the perch is the response variable. A scatterplot is provided, which depicts a linear relationship. Also included is a residual graphic for the least square regression line. For this data, the computer output is provided. As a result, it is stated,

$n=16$

$b=0.008569$

${\mathrm{SE}}_b=0.002456$

The sample size was reduced by two degrees of freedom:

$df=n-2$

$=16-2$

$=14$

In table$\mathrm{t}$, the critical $\mathrm{B}$-value can be located in the row of $df=14$and the column of $c=90\%$as

$t^{*}=1.761$

As a result, the confidence interval is as follows:

$b-t^{*}\times {\mathrm{SE}}_b$

$=0.008569-1.761\times 0.002456$

$=0.004244$

$b+t^{*}\times {\mathrm{SE}}_b$

$=0.008569+1.761\times 0.002456$

$=0.012894$

Part(c) Step 1: Given information

Given in the question that, We need to interpret the interval from part (b) in context.

Part(c) Step 2: Explanation

In order to answer the topic, researchers looked at how nature manages the size of animal predators that destroy a bigger proportion of the population when prey density is low. The amount of perch in a confined region is the explanatory variable, and the proportion of perch killed by bass in two hours while bass have access to the perch is the response variable. A scatterplot is provided, which depicts a linear relationship. Also included is a residual graphic for the least square regression line. For this data, the computer output is provided. As a result of portion (b),$(0.004244,0.012894)$

As a result, we have $90$percent confidence that the true slope of the population regression line is between $(0.004244,0.012894).$

Part(d) Step 1: Given information

Given in the question that, We need to explain the meaning of “$90\%$ confident” in context.

Part(d) Step 2: Explanation

In order to answer the topic, researchers looked at how nature manages the size of animal predators that destroy a bigger proportion of the population when prey density is low. The amount of perch in a confined region is the explanatory variable, and the proportion of perch killed by bass in two hours while bass have access to the perch is the response variable. A scatterplot is provided, which depicts a linear relationship. Also included is a residual graphic for the least square regression line. For this data, the computer output is provided. In this instance, "confident" means that the confidence interval will contain the correct value of the population parameter in $90\%$ of all available samples.