91Ó°ÊÓ

Grain Yields: Feeding the World With an ever-increasing world population, grain yields are extremely important. A random sample of 16 large grainproducing regions in the world gave the following information about grain production (in kg/hectare). (Reference: Handbook of International Economic Statistics, U.S. Government Documents.) \begin{tabular}{l|cccccccc} \hline Region & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline Modern Production & 1610 & 2230 & 5270 & 6990 & 2010 & 4560 & 780 & 6510 \\ \hline Historic Production & 1590 & 2360 & 5161 & 7170 & 1920 & 4760 & 660 & 6320 \\ \hline \end{tabular} \begin{tabular}{l|cccccccc} \hline Region & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 \\ \hline Modern Production & 2850 & 3550 & 1710 & 2050 & 2750 & 2550 & 6750 & 3670 \\ \hline Historic Production & 2920 & 2440 & 1340 & 2180 & 3110 & 2070 & 7330 & 2980 \\ \hline \end{tabular} Does this information indicate that modern grain production is higher? Use a 5\% level of significance.

Step by step solution

01

Define Hypotheses

We formulate the null and alternative hypotheses. The null hypothesis (H_0) is that there is no difference in the mean production between modern and historic production, while the alternative hypothesis (H_a) is that modern production is higher. Thus, \(H_0: \mu_{\text{modern}} = \mu_{\text{historic}}\) and \(H_a: \mu_{\text{modern}} > \mu_{\text{historic}}\).

02

Calculate Differences

For each region, calculate the difference between modern production and historic production. Calculate: 20, -130, 109, -180, 90, -200, 120, 190, -70, 1110, 370, -130, -360, 480, -580, 690.

03

Compute Test Statistic Values

First, compute the mean (\bar{x}\u00a0) and standard deviation (s) of these differences. Then use these to calculate the test statistic: \(t = \frac{\bar{x}}{s/\sqrt{n}}\).

04

Determine Critical Value

For \(n = 16\) samples, the degrees of freedom is \(n-1 = 15\). Use a t-table to find the critical t-value for a one-tailed test at a 5% level of significance, which corresponds to \(t_{0.05}\).

05

Decision

Compare the calculated t-value from Step 3 with the critical t-value from Step 4. If the calculated t-value is greater than the critical t-value, reject the null hypothesis. Otherwise, do not reject the null hypothesis.

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.

Null Hypothesis

The null hypothesis, often denoted as \( H_0 \), is a statement that there is no effect or no difference in the context of a hypothesis test. It serves as the default or starting assumption when testing a claim. In our exercise, we are looking at modern and historic grain production. The null hypothesis is that there is no significant difference between them. Mathematically, this is expressed as

• \( H_0: \mu_{\text{modern}} = \mu_{\text{historic}} \)

where \( \mu_{\text{modern}} \) and \( \mu_{\text{historic}} \) represent the mean productions for modern and historic methods, respectively.
By stating there is no difference, the null hypothesis is essentially betting on the status quo. If our analysis finds enough evidence against this claim, we may reject the null hypothesis in favor of the alternative.

Alternative Hypothesis

The alternative hypothesis, denoted as \( H_a \) or sometimes \( H_1 \), represents what we aim to support. It is the claim that we suspect might be true. In the context of our grain production exercise, the alternative hypothesis suggests that modern production is indeed higher than historic production. This is expressed as:

• \( H_a: \mu_{\text{modern}} > \mu_{\text{historic}} \)

The alternative hypothesis is crucial because it specifies the direction of the effect or difference we are testing for. In this example, because we are interested in whether modern production is higher, we use a one-tailed test. This hypothesis guides our analysis and determines the nature of our statistical test.

t-Test

The t-test is a statistical test used to determine if there is a significant difference between the means of two groups. It is particularly useful when comparing small samples, like in our exercise with only 16 regions. The test statistic for a one-sample t-test is calculated using the formula:\[ t = \frac{\bar{x}}{s/\sqrt{n}} \]where \( \bar{x} \) is the mean of the differences in production between modern and historic methods, \( s \) is the standard deviation of these differences, and \( n \) is the sample size.
The t-test helps us decide whether the observed differences in our sample are large enough to conclude that this pattern is likely to hold for the entire population. Given our hypotheses, we calculate the t-statistic and compare it to a critical value to make our decision.

Significance Level

The significance level, often denoted as \( \alpha \), is the threshold at which we decide whether a result is statistically significant. It represents the probability of rejecting the null hypothesis when it is actually true. In many studies, a common significance level used is 5%, or 0.05.

• In this exercise, our significance level is set at 5%. This means we only have a 5% risk of concluding that modern production is higher when it's not.

To make a decision based on our t-test, we compare the p-value or the test statistic to the critical value associated with \( \alpha = 0.05 \). If our result falls into the critical region, where the probability of it being due to chance is <= 5%, we will reject the null hypothesis. This is a key part of hypothesis testing, ensuring that our conclusions are made with a defined level of confidence.