91Ó°ÊÓ

The article "Effect of Temperature on the pH of Skim Milk" (Journal of Dairy Research [1988]: 277- 280) reported on a study involving \(x=\) temperature \(\left({ }^{\circ} \mathrm{C}\right)\) under specified experimental conditions and \(y=\) milk \(\mathrm{pH}\). The accompanying data (read from a graph) are a representative subset of that which appeared in the article: \(\begin{array}{rrrrrrrrr}x & 4 & 4 & 24 & 24 & 25 & 38 & 38 & 40 \\ y & 6.85 & 6.79 & 6.63 & 6.65 & 6.72 & 6.62 & 6.57 & 6.52\end{array}\) $$ \begin{array}{lrrrrrrrr} x & 45 & 50 & 55 & 56 & 60 & 67 & 70 & 78 \\ y & 6.50 & 6.48 & 6.42 & 6.41 & 6.38 & 6.34 & 6.32 & 6.34 \\ \sum x=678 & \sum y=104.54 & \sum x^{2}=36,056 & \\ \sum y^{2}=683.4470 & & \sum x y=4376.36 & & \end{array} $$ Do these data strongly suggest that there is a negative linear relationship between temperature and \(\mathrm{pH}\) ? State and test the relevant hypotheses using a significance level of \(.01\).

Step by step solution

01

Calculate Mean values for X and Y

The mean values are given by the formula: \[\overline{x}=\sum x/n \text{ and } \overline{y}=\sum y/n\] where \(\overline{x}\) and \(\overline{y}\) are the means of X and Y respectively and n is the number of observations for each variable. The sum of X, ∑X=678, and the sum of Y, ∑Y=104.54, are given. The number of observations can be counted, which amounts to 17. Using these values, calculate the mean values.

02

Calculate the Pearson correlation coefficient (r)

The Pearson correlation coefficient (r) can be calculated using this formula: \[ r = \frac{{n\left(\sum xy\right) - \left(\sum x\right)\left(\sum y\right)}}{{\sqrt{\left[n\sum x^2 - \left(\sum x\right)^2\right]\left[n\sum y^2 - \left(\sum y\right)^2\right]}}}\] where n is the number of observations, ∑xy is the sum of the product of X and Y, ∑x and ∑y are the sums of X and Y respectively, and ∑x² and ∑y² are the sums of squares of X and Y respectively. Substituting the values, ∑xy=4376.36, ∑x²=36056, ∑y²=683.4470 and the just calculated mean values from step 1, compute the Pearson coefficient.

03

Test the Hypothesis

The null (H0) and alternative (Ha) hypotheses are as follows: H0: r = 0 (No linear relationship) Ha: r ≠ 0 (there's a linear relationship). To test the null hypothesis against the alternative, using a significance level of 0.01 and 15 degrees of freedom (n-2), compare the calculated Pearson coefficient from step 2 to the critical t-value from the t-distribution table.

04

Conclusion

If the calculated Pearson coefficient is less than the negative critical value or more than the positive critical value, reject the null hypothesis and conclude that there is a significant negative linear relationship. If it is not, fail to reject null hypothesis and conclude that the evidence is not strong enough to assert a linear relationship.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Pearson correlation coefficient

The Pearson correlation coefficient is a statistical measure that quantifies the degree to which two variables are linearly related. In essence, it helps us understand if changes in one variable tend to be associated with proportional changes in another. The values of the Pearson coefficient, often denoted as \(r\), range from -1 to 1.

• A Pearson coefficient of 1 indicates a perfect positive linear relationship, meaning as one variable increases, the other variable increases proportionally.
• A value of -1 signifies a perfect negative linear relationship, which means that as one variable increases, the other decreases proportionally.
• When the Pearson coefficient is 0, it suggests no linear relationship between the variables.

In the context of the given study on temperature and milk pH, the Pearson coefficient helps us determine if there is a linear association between these two factors. Using the formula for \(r\), researchers substitute the sums of the variables and their products to compute this coefficient. If the computed \(r\) is significantly different from zero, it implies that temperature might have a tangible effect on milk pH. Calculating this value with accuracy is crucial for drawing meaningful conclusions about the relationship.

Hypothesis testing

Hypothesis testing is a statistical process used to determine whether a given hypothesis about a data set is supported by the data. It usually involves comparing observed data with a specific hypothesis and using a significance level, which is the probability of rejecting the null hypothesis when it is true. The given exercise centers on testing the hypothesis about the linear relationship between temperature and milk pH. Here are the two main hypotheses:

• Null Hypothesis \((H_0)\): This claims that there is no linear relationship between temperature and milk pH, expressed as \(r = 0\).
• Alternative Hypothesis \((H_a)\): This asserts that there is a linear relationship, which means \(r eq 0\).

With a level of significance set at 0.01, we check if the observed correlation could have been a result of random chance. If the calculated Pearson correlation coefficient leads us to reject the \(H_0\), then the evidence points to a legitimate linear connection between the two variables.

Linear relationship

A linear relationship is a statistical term used to describe a relationship between two variables that can be represented with a straight line when plotted on a graph. In a linear relationship, any change in the independent variable \(x\) results in a proportional change in the dependent variable \(y\). In the context of the experiment, a linear relationship between temperature and milk pH suggests that as the temperature increases, the pH of milk changes predictably in a specific direction. Based on the data set provided, you can deduce whether a positive or negative slope exists.

• If the relationship is negative, as the temperature rises, the pH decreases, forming a downward-sloping line.
• If it were a positive slope (hypothetically), both variables would increase together.

Linear relationships are fundamental in predicting outcomes and understanding dependencies between variables in statistics. Here, if a strong negative linear relationship is found, it indicates that managing temperature could help in controlling the acidity levels in milk.