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An article in the Journal of Agricultural Science [ "The Use of Residual Maximum Likelihood to Model Grain Quality Characteristics of Wheat with Variety, Climatic and Nitrogen Fertilizer Effects" (1997, Vol. 128, pp. \(135-142\) ) ] investigated means of wheat grain crude protein content (CP) and Hagberg falling number (HFN) surveyed in the UK. The analysis used a variety of nitrogen fertilizer applications (kg N/ha), temperature \(\left({ }^{\circ} \mathrm{C}\right),\) and total monthly rainfall \((\mathrm{mm})\). The data shown below describe temperatures for wheat grown at Harper Adams Agricultural College between 1982 and \(1993 .\) The temperatures measured in June were obtained as follows: $$ \begin{array}{llllll} 15.2 & 14.2 & 14.0 & 12.2 & 14.4 & 12.5 \\ 14.3 & 14.2 & 13.5 & 11.8 & 15.2 & \end{array} $$ Assume that the standard deviation is known to be \(\sigma=0.5\). (a) Construct a \(99 \%\) two-sided confidence interval on the mean temperature. (b) Construct a \(95 \%\) lower-confidence bound on the mean temperature. (c) Suppose that we wanted to be \(95 \%\) confident that the error in estimating the mean temperature is less than 2 degrees Celsius. What sample size should be used? (d) Suppose that we wanted the total width of the two-sided confidence interval on mean temperature to be 1.5 degrees Celsius at \(95 \%\) confidence. What sample size should be used?

Step by step solution

01

Calculate Sample Mean

First, we need to calculate the sample mean of the given temperatures. The data is: \[ 15.2, 14.2, 14.0, 12.2, 14.4, 12.5, 14.3, 14.2, 13.5, 11.8, 15.2 \]To find the mean, sum all the temperatures and divide by the number of observations (11):\[ \bar{x} = \frac{15.2 + 14.2 + 14.0 + 12.2 + 14.4 + 12.5 + 14.3 + 14.2 + 13.5 + 11.8 + 15.2}{11} \]\[ \bar{x} \approx 13.85 \]

02

Construct 99% Confidence Interval

To construct a 99% confidence interval for the mean, we use the formula:\[ \bar{x} \pm Z_{\alpha/2} \frac{\sigma}{\sqrt{n}} \]where \(Z_{\alpha/2} = 2.576\) for 99% confidence level, \(\sigma = 0.5\), and \(n = 11\).The margin of error is computed as:\[ 2.576 \times \frac{0.5}{\sqrt{11}} \approx 0.388 \]So, the confidence interval is:\[ 13.85 \pm 0.388 \]\[ (13.462, 14.238) \]

03

Construct 95% Lower Confidence Bound

For a 95% lower confidence bound, we use:\[ \bar{x} - Z_{\alpha} \frac{\sigma}{\sqrt{n}} \]where \(Z_{\alpha} = 1.645\) for 95% confidence levelThe bound is computed as:\[ 13.85 - 1.645 \times \frac{0.5}{\sqrt{11}} \approx 13.61 \]So, the lower confidence bound is:\[ 13.61 \]

04

Determine Sample Size for 95% Confidence with Error < 2°C

We are given an error margin of less than 2°C. Using:\[ E = Z_{\alpha/2} \frac{\sigma}{\sqrt{n}} \leq 2 \]For 95% confidence, \(Z_{\alpha/2} = 1.96\). Substituting, we solve for \(n\):\[ 1.96 \times \frac{0.5}{\sqrt{n}} = 2 \]Rearranging for \(n\):\[ n = \left( \frac{1.96 \times 0.5}{2} \right)^2 \]\[ n = \left( \frac{0.98}{2} \right)^2 = 0.24^2 \approx 0.0576 \]Thus, the minimum \(n\) is 1, but practically \(n\) must be greater for error constraints.

05

Determine Sample Size for Confidence Interval Width 1.5°C at 95% Confidence

We want the total width of the confidence interval to be 1.5°C:\[ 2 \times Z_{\alpha/2} \frac{\sigma}{\sqrt{n}} = 1.5 \]With \(Z_{\alpha/2} = 1.96\) for 95% confidence:\[ 2 \times 1.96 \times \frac{0.5}{\sqrt{n}} = 1.5 \]Then solve for \(n\):\[ n = \left( \frac{2 \times 1.96 \times 0.5}{1.5} \right)^2 \]\[ n = \left( \frac{1.96}{1.5} \right)^2 \]\[ n = \left( 1.3 \right)^2 \approx 1.69 \]\[ n \approx 2.9 \]Round up, the minimum sample size \(n\) is 3.

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Key Concepts

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

Sample Size Calculation

Calculating the right sample size is crucial in statistical analysis. It determines how accurately your results will reflect the true population parameters. In our exercise, we need to calculate the sample size for two scenarios: keeping the margin of error less than 2°C and having a confidence interval width of 1.5°C. The formula used is similar for both but the objective differs slightly.

Here's the formula you'll often see:

• To keep the margin of error under a set value: \[ n = \left( \frac{Z_{\alpha/2} \times \sigma}{E} \right)^2 \]
• For desired confidence interval width: \[ n = \left( \frac{2 \times Z_{\alpha/2} \times \sigma}{\text{width}} \right)^2 \]

Variables to know:

• \( n \): the sample size required
• \( Z_{\alpha/2} \): Z-score corresponding to the confidence level
• \( \sigma \): known standard deviation
• \( E \): acceptable error, or margin of error
• \( \text{width} \): the total acceptable width for the confidence interval

Sampling accurately ensures that your study results are reliable. For example, in the case of wanting a CI width of 1.5°C, rounding calculations suggest we need at least 3 more samples for assurance.

Z-score

The Z-score is a vital concept in statistics, often used to determine the standard deviations a data point is from the mean. It plays a crucial role in calculating confidence intervals and margins of error. In our exercise, the Z-score helps us find the critical value needed for constructing confidence intervals and determining sample sizes.

For confidence intervals:

• The Z-score (\( Z_{\alpha/2} \)) signifies the number of standard deviations needed to capture the central portion of the normal distribution, based on the confidence level:
• For 95% confidence, \( Z_{\alpha/2} \approx 1.96 \)
• For 99% confidence, \( Z_{\alpha/2} \approx 2.576 \)

These values are derived from standard normal distribution tables or using statistical software. Understanding Z-scores is essential as it influences the width of your confidence interval and impacts your decision-making, especially when sample sizes are limited.

Margin of Error

The margin of error represents the range that your population parameter will likely fall within your sample estimate. It's important for understanding accuracy in statistics, especially when reporting results like means or proportions. In our scenario, margin of error was computed to determine confidence levels and guide sample size decisions.

The formula for calculating the margin of error is:

• \[ \text{Margin of Error} = Z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}} \]

What each part means:

• \( \sigma \): known standard deviation — a direct indicator of dataset variability
• \( n \): sample size — larger samples reduce the margin of error
• \( Z_{\alpha/2} \): Z-score — reflects the level of confidence

In practice, a smaller margin of error indicates more precise estimates. However, achieving this typically requires larger sample sizes, hence the exercises on determining adequate sample sizes to meet specific conditions.