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Chapter 10: Problem 53

State the null and alternative hypotheses that would be used to test the following claims: a. There is a difference between the mean age of employees at two different large companies. b. The mean of population 1 is greater than the mean of population 2 c. The mean yield of sunflower seeds per county in North Dakota is less than the mean yield per county in South Dakota. d. There is no difference in the mean number of hours spent studying per week between male and female college students.

Short Answer

Expert verified
For claim a, \(H_0: \mu_1 - \mu_2 = 0\) and \(H_a: \mu_1 - \mu_2 \neq 0\). For claim b, \(H_0: \mu_1 \leq \mu_2\) and \(H_a: \mu_1 > \mu_2\). For claim c, \(H_0: \mu_{ND} \geq \mu_{SD}\) and \(H_a: \mu_{ND} < \mu_{SD}\). For claim d, \(H_0: \mu_{M} = \mu_{F}\) and \(H_a: \mu_{M} \neq \mu_{F}\).

Step by step solution

01

Hypothesis for the difference between means of two companies

For claim a, the null hypothesis \(H_0\) would be that the difference between the mean ages of employees at the two companies is zero (i.e., there is no difference), thus \(H_0: \mu_1 - \mu_2 = 0\). The alternative hypothesis \(H_a\) would be that there is a difference, thus \(H_a: \mu_1 - \mu_2 \neq 0\).
02

Hypothesis for mean of population 1 is greater than population 2

For claim b, the null hypothesis \(H_0\) would be that the mean of population 1 is not greater than the mean of population 2, thus \(H_0: \mu_1 \leq \mu_2\). The alternative hypothesis \(H_a\) is the opposite of the null, thus \(H_a: \mu_1 > \mu_2\).
03

Hypothesis for mean yield of sunflower seeds per county

For claim c, the null hypothesis \(H_0\) is that the mean yield of sunflower seeds per county in North Dakota is not less than South Dakota, thus \(H_0: \mu_{ND} \geq \mu_{SD}\). The alternative hypothesis \(H_a\) would be that the mean yield in North Dakota is less than South Dakota, thus \(H_a: \mu_{ND} < \mu_{SD}\).
04

Hypothesis for mean number of studying hours between genders

For claim d, the null hypothesis \(H_0\) is that there's no difference in mean studying hours between male and female students, thus \(H_0: \mu_{M} = \mu_{F}\). The alternative hypothesis \(H_a\) would be that there is a difference, thus \(H_a: \mu_{M} \neq \mu_{F}\).

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Key Concepts

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

Null and Alternative Hypotheses
Understanding the null hypothesis (\( H_0 \) and the alternative hypothesis (\( H_a \) is crucial when it comes to hypothesis testing in statistics. The null hypothesis is a statement that implies no effect or no difference in the context of the research question. It is the hypothesis that one seeks to test and, typically, to nullify or disprove. For example, if we're looking at whether there's a difference between two groups' means, the null hypothesis would generally state that the difference between the groups is zero.

On the contrary, the alternative hypothesis (\( H_a \) represents what a researcher wants to prove. This hypothesis suggests that there is a statistically significant effect or difference. The direction of the alternative hypothesis can be two-sided (meaning there could be any difference), greater than, or less than, depending on the research question. Referring to our textbook problems, for a claim that the mean age of employees at two companies differs, the alternative hypothesis challenges the null by suggesting a specific difference exists between their mean ages.
Mean Comparison
When comparing means between two groups, statisticians use a variety of tests to determine if the observed differences are statistically significant or if they might have occurred by chance. This process is typically conducted by calculating the means of each group and then comparing them using a formula that accounts for the variability within the groups and the size of the groups.

If we're comparing the mean yield of sunflower seeds per county between North Dakota and South Dakota, we're essentially checking if the average yield from one state is significantly different from the other. In this scenario, one would calculate the mean yield for each state individually and then determine if any observed difference is large enough to be unlikely to have arisen under the assumption that our null hypothesis of no difference is true.
Statistical Significance
Statistical significance is a term used to describe whether the results of an experiment or study are likely to be true and not occurred by chance. This concept relates to the probability that the observed effect or difference would occur if the null hypothesis were true. In hypothesis testing, this probability is known as the p-value. A low p-value (usually less than 0.05) indicates that the null hypothesis can be rejected, meaning that the findings are statistically significant and not due to random chance.

Importance of P-Value:

When determining the statistical significance, the p-value helps us make a decision about our null hypothesis. For instance, when investigating whether studying hours differ between genders, if the resulting p-value is less than our significance level (let's say 0.05), we then have evidence to reject our null hypothesis and accept the alternative hypothesis, suggesting a real difference exists.

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Most popular questions from this chapter

If a random sample of 18 homes south of Center Street in Provo has a mean selling price of \(\$ 145,200\) and a standard deviation of \(\$ 4700,\) and a random sample of 18 homes north of Center Street has a mean selling price of \(\$ 148,600\) and a standard deviation of \(\$ 5800,\) can you conclude that there is a significant difference between the selling prices of homes in these two areas of Provo at the 0.05 level? Assume normality. a. Solve using the \(p\) -value approach. b. Solve using the classical approach.

An experiment is designed to study the effect diet has on uric acid level. The study includes 20 white rats. Ten rats are randomly selected and given a junk- food diet; the other 10 rats receive a high-fiber, low-fat diet. Uric acid levels of the two groups are determined. Do the resulting sets of data represent dependent or independent samples? Explain.

One could reason that high school seniors would have more money issues than high school juniors. Seniors foresee expenses for college as well as their senior trip and prom. So does this mean that they work more than their junior classmates? Christine, a senior at HFL High School, randomly collected the following data (recorded in hours/week) from students that work: $$\begin{array}{llll}\hline \text { Seniors } & n_{s}=17 & \bar{x}_{s}=16.4 & s_{s}=10.48 \\\\\text { Juniors } & n_{i}=20 & \bar{x}_{i}=18.405 & s_{i}=9.69 \\\\\hline\end{array}$$ Assuming that work hours are normally distributed, do these data suggest that there is a significant difference between the average number of hours that HFL seniors and juniors work per week? Use \(\alpha=0.10\).

Two independent random samples of size 25 were taken from an English class and a chemistry class at a local community college. Students in both classes were asked to draw a 3 -inch line to the best of their ability without any measuring device (ruler, etc.). The following data resulted: $$\begin{array}{llll}\hline \text { English } & n=25 & \bar{x}=2.660 & s=0.617 \\\\\text { Chemistry } & \mathrm{n}=25 & \bar{x}=2.750 & \mathrm{s}=0.522 \\\\\hline\end{array}$$ At the 0.05 level of significance, is there a difference between the standard deviations for 3 -inch-line measurements from the English and chemistry classes?

A bakery is considering buying one of two gas ovens. The bakery requires that the temperature remain constant during a baking operation. A study was conducted to measure the variance in temperature of the ovens during the baking process. The variance in temperature before the thermostat restarted the flame for the Monarch oven was 2.4 for 16 measurements. The variance for the Kraft oven was 3.2 for 12 measurements. Does this information provide sufficient reason to conclude that there is a difference in the variances for the two ovens? Assume measurements are normally distributed and use a 0.02 level of significance.
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