91Ó°ÊÓ

According to the size of a female salamander's snout is correlated with the number of eggs in her clutch. The following data are consistent with summary quantities reported in the article. Partial Minitab output is also included. \(\begin{array}{lrrrrr}\text { Snout-Vent Length } & 32 & 53 & 53 & 53 & 54 \\\ \text { Clutch Size } & 45 & 215 & 160 & 170 & 190 \\ \text { Snout-Vent Length } & 57 & 57 & 58 & 58 & 59 \\ \text { Clutch Size } & 200 & 270 & 175 & 245 & 215 \\ \text { Snout-Vent Length } & 63 & 63 & 64 & 67 & \\ \text { Clutch Size } & 170 & 240 & 245 & 280 & \end{array}\) 2 The regression equation is \(\begin{aligned}&Y=-133+5.92 x \\\&\text { Predictor } & \text { Coef } & \text { StDev } & T & P \\\&\text { Constant } & -133.02 & 64.30 & 2.07 & 0.061 \\\&x & 5.919 & 1.127 & 5.25 & 0.000 \\\&s=33.90 & \text { R-Sq }=69.7 \% & R-S q(a d j)=67.2 \%\end{aligned}\) Additional summary statistics are \(n=14 \quad \bar{x}=56.5 \quad \bar{y}=201.4\) \(\sum x^{2}=45,958 \quad \sum y^{2}=613,550 \quad \sum x y=164,969\) a. What is the equation of the regression line for predicting clutch size based on snout-vent length? b. What is the value of the estimated standard deviation of \(b\) ? c. Is there sufficient evidence to conclude that the slope of the population line is positive? d. Predict the clutch size for a salamander with a snoutvent length of 65 using a \(95 \%\) interval. e. Predict the clutch size for a salamander with a snoutvent length of 105 using a \(90 \%\) interval.

Step by step solution

01

Equation of the Regression Line

The equation of the regression line, also known as the best fit line, is given by \(Y = -133 + 5.92x\). Therefore, this is the equation for predicting clutch size based on snout-vent length. In this equation, \(x\) represents the snout-vent length, and \(Y\) represents the predicted clutch size.

02

Value of Estimated Standard Deviation

The estimated standard deviation of 'b' (slope), denoted as \(s_b\), is the standard error of the regression slope. Looking at the given summary statistics, we see for x (snout-vent length) the standard deviation value is 1.127.

03

Slope of the Population Line

To assess whether there is sufficient evidence to conclude that the slope of the population line is positive, we look at the P-value associated with the slope. The P-value of x is given as 0.000. This P-value is less than the common alpha level of 0.05, so we can reject the null hypothesis that the slope is zero (or negative). This provides sufficient evidence to conclude that the slope of the population line is positive.

04

Predicting Clutch Size (65 snout-vent length)

To predict the clutch size for a salamander with a snout-vent length of 65 using a 95% interval, we substitute \(x = 65\) into the regression equation: \(Y = -133 + 5.92(65) = 251.4\). The 95% interval calculation would involve t-statistics and standard error, which are not provided, so cannot be obtained here.

05

Predicting Clutch Size (105 snout-vent length)

Similarly, to predict the clutch size for a salamander with a snout-vent length of 105 using a 90% interval, we substitute \(x = 105\) into the regression equation: \(Y = -133 + 5.92(105) = 489.6\). The 90% interval calculation would involve t-statistics and standard error, which are not provided, so cannot be obtained here.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Predictor Variables

When it comes to understanding regression analysis, the term predictor variables plays a key role. These are the independent variables in a regression equation that are used to predict the value of the dependent variable. In our exercise, the snout-vent length of a salamander is the predictor variable because it is used to anticipate the clutch size, which is the dependent variable.

In regression analysis, the relationship between the predictor and the outcome is quantified by the regression coefficients. These coefficients tell us the amount of change we can expect in the dependent variable for a one-unit change in the predictor variable, assuming all other variables remain constant.

Standard Deviation

The standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean of the set, while a high standard deviation indicates that the values are spread out over a wider range. In the context of the exercise, the standard deviation value of 1.127 for the predictor variable's coefficient signifies the dispersion of snout-vent length values around the estimated regression line.

Understanding standard deviation in the scope of regression helps us assess the reliability of the predictor. If the standard deviation of the estimated coefficient is high, the reliability of the prediction decreases since there's more variability in the data points.

Null Hypothesis

The null hypothesis is a fundamental concept in hypothesis testing used as a starting point to test if there is a significant effect or relationship between variables. It proposes that there is no effect or no difference, and any observed effect is due to random chance.

In regression analysis, the null hypothesis often states that there is no relationship between the predictor and dependent variable – meaning, the regression coefficient is zero. This hypothesis is tested against an alternative hypothesis, which proposes that there is indeed an effect or relationship. Rejecting the null hypothesis implies that there is statistically significant evidence to support the existence of a relationship.

P-value

The P-value is the probability of obtaining test results at least as extreme as the observed data, assuming that the null hypothesis is true. In simpler terms, it's a measure used to determine the statistical significance of the findings from a study or an experiment.

In the salamander exercise, the P-value associated with the slope is 0.000, which is less than the commonly used significance level of 0.05. This extremely low P-value indicates that the probability of finding such a result, if in fact, the true slope is zero, is virtually non-existent. Therefore, we have sufficient evidence to reject the null hypothesis and conclude that there is a positive relationship between snout-vent length and clutch size.

Confidence Interval

A confidence interval is a range of values that's used to estimate the true value of an unknown population parameter. The confidence level represents the proportion of times that the interval estimate would contain the parameter if you repeated the study multiple times.

For instance, a 95% confidence interval means that if we repeated the study 100 times, we would expect the interval to contain the true parameter 95 times out of 100. It's important to note that the confidence interval includes an element of uncertainty. In regression, confidence intervals for predictions account for both the uncertainty in estimating the population mean and the random variation around this mean.