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You are a keen gardener and keep a great array of houseplants in your dorm. You are thinking about trying to water them less often because you are worried that you are over-watering, so you carry out an experiment. You randomly choose 12 of your plants and assign them to group 1, you randomly pick another 7 plants and then assign them to group 2. The average growth after 2 weeks for group 1 with the old amount of water is 3cm with a standard deviation of 0.8 cm. For group 2, the group with the new amount of water, the average growth rate is 3.4cm with a standard deviation of 0.45cm. Let μ₁ be the true growth rate under the old amount of watering and µ2 be the true growth rate under the new amount of watering. Are these two means different? Perform a test at an a = 0.01 level. (a) Set up the null and alternative hypothesis (using mathematical notation/numbers AND interpret them in context of the problem). (b) Calculate the test statistic. (c) Calculate the critical value. (d) Draw a picture of the distribution of the test statistic under Ho. Label and provide values for the critical value and the test statistic, and shade the critical region.

You are a keen gardener and keep a great array of houseplants in your dorm. You are thinking about trying to water them less often because you are worried that you are over-watering, so you carry out an experiment. You randomly choose 12 of your plants and assign them to group 1, you randomly pick another 7 plants and then assign them to group 2. The average growth after 2 weeks for group 1 with the old amount of water is 3cm with a standard deviation of 0.8 cm. For group 2, the group with the new amount of water, the average growth rate is 3.4cm with a standard deviation of 0.45cm. Let μ₁ be the true growth rate under the old amount of watering and µ2 be the true growth rate under the new amount of watering. Are these two means different? Perform a test at an a = 0.01 level. (a) Set up the null and alternative hypothesis (using mathematical notation/numbers AND interpret them in context of the problem). (b) Calculate the test statistic. (c) Calculate the critical value. (d) Draw a picture of the distribution of the test statistic under Ho. Label and provide values for the critical value and the test statistic, and shade the critical region.

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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**Title: Statistical Analysis on the Impact of Watering Amount on Plant Growth** **Introduction:** As a keen gardener, you always strive to maintain the perfect environment for your houseplants in your dorm. Concerned about potentially over-watering your plants, you decide to experiment with reducing the watering frequency. By employing statistical tests, you aim to determine whether the change in watering amount significantly affects plant growth. **Experiment Setup:** You select 12 plants and assign them to Group 1. Then you select another 7 plants and assign them to Group 2. Over a span of two weeks, both groups receive different watering treatments, after which the following growth data is recorded: - **Group 1 (Old Amount of Watering):** - Average Growth: 3 cm - Standard Deviation: 0.8 cm - Sample Size (\(n_1\)): 12 - **Group 2 (New Amount of Watering):** - Average Growth: 3.4 cm - Standard Deviation: 0.45 cm - Sample Size (\(n_2\)): 7 The goal is to test whether there is a significant difference in the average growth rates between the two groups at a significance level (\(\alpha\)) of 0.01. **Hypothesis Testing:** **(a) Null and Alternative Hypotheses:** Set up the null and alternative hypotheses in mathematical notation and interpret them in the context of the problem: - \(H_0\): \(\mu_1 = \mu_2\) (The mean growth rate under the old amount of watering is equal to the mean growth rate under the new amount of watering.) - \(H_a\): \(\mu_1 \neq \mu_2\) (The mean growth rate under the old amount of watering is not equal to the mean growth rate under the new amount of watering.) **(b) Calculate the Test Statistic:** Given the data, calculate the test statistic for the comparison of the two means. **(c) Calculate the Critical Value:** Using the significance level \(\alpha = 0.01\), find the critical value for the test statistic. **(d) Visualization:** Draw the distribution of the test statistic under the null hypothesis, \(H_0\). Label and provide values for the critical value and the test statistic, and shade the critical region.
Transcribed Image Text:**Title: Statistical Analysis on the Impact of Watering Amount on Plant Growth** **Introduction:** As a keen gardener, you always strive to maintain the perfect environment for your houseplants in your dorm. Concerned about potentially over-watering your plants, you decide to experiment with reducing the watering frequency. By employing statistical tests, you aim to determine whether the change in watering amount significantly affects plant growth. **Experiment Setup:** You select 12 plants and assign them to Group 1. Then you select another 7 plants and assign them to Group 2. Over a span of two weeks, both groups receive different watering treatments, after which the following growth data is recorded: - **Group 1 (Old Amount of Watering):** - Average Growth: 3 cm - Standard Deviation: 0.8 cm - Sample Size (\(n_1\)): 12 - **Group 2 (New Amount of Watering):** - Average Growth: 3.4 cm - Standard Deviation: 0.45 cm - Sample Size (\(n_2\)): 7 The goal is to test whether there is a significant difference in the average growth rates between the two groups at a significance level (\(\alpha\)) of 0.01. **Hypothesis Testing:** **(a) Null and Alternative Hypotheses:** Set up the null and alternative hypotheses in mathematical notation and interpret them in the context of the problem: - \(H_0\): \(\mu_1 = \mu_2\) (The mean growth rate under the old amount of watering is equal to the mean growth rate under the new amount of watering.) - \(H_a\): \(\mu_1 \neq \mu_2\) (The mean growth rate under the old amount of watering is not equal to the mean growth rate under the new amount of watering.) **(b) Calculate the Test Statistic:** Given the data, calculate the test statistic for the comparison of the two means. **(c) Calculate the Critical Value:** Using the significance level \(\alpha = 0.01\), find the critical value for the test statistic. **(d) Visualization:** Draw the distribution of the test statistic under the null hypothesis, \(H_0\). Label and provide values for the critical value and the test statistic, and shade the critical region.
### Statistical Decision-Making and Confidence Intervals #### Problem Evaluation and Decision Justification **(e)** Make and justify a statistical decision at the α = 0.01 level and state your conclusions in context of the problem. This question prompts students to perform a statistical hypothesis test at a significance level of α = 0.01. Based on the results, you should decide whether to reject or fail to reject the null hypothesis and explain your decision within the context of the given problem. #### Constructing Confidence Intervals **(f)** Construct a 99% confidence interval for the true difference in growth rates from the 2 different fertilizers. In this step, students are required to construct a confidence interval for the difference in growth rates between two types of fertilizers. A 99% confidence interval will provide a range of values for this difference with 99% confidence. #### Reaffirmation of Results **(g)** Does this interval reaffirm your statistical decision from the hypothesis test? Explain. This question asks you to interpret the confidence interval constructed in part (f) and compare it to your hypothesis test conclusions from part (e). If the interval suggests a significant difference (e.g., does not include zero), it should reaffirm your decision from the hypothesis test. Make sure to provide detailed explanations and calculations to support your conclusions in each part. Explore how the context of the problem influences your decision and interpretation of results.
Transcribed Image Text:### Statistical Decision-Making and Confidence Intervals #### Problem Evaluation and Decision Justification **(e)** Make and justify a statistical decision at the α = 0.01 level and state your conclusions in context of the problem. This question prompts students to perform a statistical hypothesis test at a significance level of α = 0.01. Based on the results, you should decide whether to reject or fail to reject the null hypothesis and explain your decision within the context of the given problem. #### Constructing Confidence Intervals **(f)** Construct a 99% confidence interval for the true difference in growth rates from the 2 different fertilizers. In this step, students are required to construct a confidence interval for the difference in growth rates between two types of fertilizers. A 99% confidence interval will provide a range of values for this difference with 99% confidence. #### Reaffirmation of Results **(g)** Does this interval reaffirm your statistical decision from the hypothesis test? Explain. This question asks you to interpret the confidence interval constructed in part (f) and compare it to your hypothesis test conclusions from part (e). If the interval suggests a significant difference (e.g., does not include zero), it should reaffirm your decision from the hypothesis test. Make sure to provide detailed explanations and calculations to support your conclusions in each part. Explore how the context of the problem influences your decision and interpretation of results.
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