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An article in Growth: A Journal Devoted to Problems of Normal and Abnormal Growth ["Comparison of Measured and Estimated Fat-Free Weight, Fat, Potassium and Nitrogen of Growing Guinea Pigs" \((1982,\) Vol. \(46(4),\) pp. \(306-321)]\) reported the results of a study that measured the body weight (in grams) for guinea pigs at birth. $$\begin{array}{rrrrrr}421.0 & 452.6 & 456.1 & 494.6 & 373.8 \\\90.5 & 110.7 & 96.4 & 81.7 & 102.4 \\\241.0 & 296.0 & 317.0 & 290.9 & 256.5 \\\447.8 & 687.6 & 705.7 & 879.0 & 88.8 \\\296.0 & 273.0 & 268.0 & 227.5 & 279.3 \\\258.5 & 296.0 & & &\end{array}$$ (a) Test the hypothesis that mean body weight is 300 grams. Use \(\alpha=0.05\) (b) What is the smallest level of significance at which you would be willing to reject the null hypothesis? (c) Explain how you could answer the question in part (a) with a two-sided confidence interval on mean body weight.

Step by step solution

01

Compute Sample Mean and Standard Deviation

First, combine all measurements into a single sample and find the mean \( \bar{x} \) and standard deviation \( s \). The sample mean can be computed as: \( \bar{x} = \frac{\sum x_i}{n} \), where \( n \) is the total number of measurements. After that, calculate the standard deviation using the formula \( s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} \).

02

State Hypotheses

Formulate the null hypothesis \( H_0 \) and alternative hypothesis \( H_a \). For this problem: \( H_0: \mu = 300 \) grams (the mean body weight is 300 grams), and \( H_a: \mu eq 300 \) grams (the mean body weight is not 300 grams).

03

Calculate Test Statistic

Use the t-test to calculate the test statistic \( t \) using the formula: \( t = \frac{\bar{x} - \mu}{s/\sqrt{n}} \), where \( \mu \) is the hypothesized population mean, \( \bar{x} \) is the sample mean, \( s \) is the sample standard deviation, and \( n \) is the sample size.

04

Determine Critical Value and Decision

For \( \alpha = 0.05 \) and \( n - 1 \) degrees of freedom, find the critical t-values for a two-tailed test in a t-distribution table. Compare the test statistic \( t \) to the critical values to determine if you reject or fail to reject \( H_0 \).

05

Calculate P-Value

Calculate the p-value corresponding to the computed t-statistic. Use statistical software or a t-distribution table to find this value. The p-value helps determine the smallest level of significance at which \( H_0 \) can be rejected.

06

Construct Confidence Interval

Compute a 95% confidence interval for the mean \( \mu \) using the formula \( \bar{x} \pm t_{\alpha/2} \times \frac{s}{\sqrt{n}} \). This interval helps verify the hypothesis test by checking if 300 grams lies within the interval.

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Key Concepts

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

Sample Mean

Let's begin with understanding the concept of sample mean, which is often denoted as \( \bar{x} \). The sample mean is a measure that provides an average of a data set. Basically, it sums up all the data points and divides the result by the number of points in the sample. Why is it important? Because it gives a simple snapshot of our data's central tendency.
Here's how you calculate it:

• Add up all the measurement values: \( x_i \)
• Count the number of measurements \( n \)
• Divide the total sum by \( n \): \( \bar{x} = \frac{\sum x_i}{n} \)

In practice, oftentimes you would first glimpse at the mean to sense whether your other calculations and observations are in tune with your expectations. It can act as a preliminary overview of what your full dataset might reveal.

Standard Deviation

Standard deviation is like the pulse rate of our data set, telling us how spread out the numbers are from the mean. When we talk about variability, standard deviation comes into play as a crucial statistic. If the data points are close to the mean, the standard deviation is small. Conversely, if they are spread out over a wider range, it's large.To calculate the standard deviation \( s \):

• Subtract the sample mean \( \bar{x} \) from each data point \( x_i \) to find the deviation of each entry.
• Square these deviations.
• Sum all squared deviations.
• Divide by \( n-1 \) (where \( n \) is the number of data points) - this gives the variance.
• The standard deviation is simply the square root of this variance: \( s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} \)

A key takeaway is understanding how standard deviation relates to the consistency of your dataset. High standard deviation means more variability in measurements, which may suggest the need for a larger sample size for more accurate results.

t-Test

A t-test is a statistical test used to evaluate the hypotheses when sample sizes are small and the population variance is unknown. It's particularly useful when dealing with small datasets, which are common in biological sciences and experiments like the guinea pig weight measurements mentioned.Conducting a t-test involves:

• Establishing the null hypothesis \( H_0 \) (e.g., mean equals a known value) against an alternative hypothesis \( H_a \) (e.g., mean does not equal that value).
• Calculating the test statistic \( t \) with the formula: \( t = \frac{\bar{x} - \mu}{s / \sqrt{n}} \), where \( \mu \) is the population mean under the null hypothesis.
• Finding the critical t-value from a t-distribution table corresponding with your chosen \( \alpha \) level (e.g., 0.05) and degrees of freedom \( n-1 \).

After you compute the t-value, you compare it with the critical t-value to determine whether to reject the null hypothesis. It's a way of making a binary decision based on your data, lending statistical rigor to your conclusions.

Confidence Interval

A confidence interval gives us a range of values within which we expect our true population parameter (like the mean) to lie, with a specified level of confidence (commonly 95%). It complements the hypothesis testing by providing an interval estimate as opposed to a single point estimate.To calculate a confidence interval around the mean, you:

• Start with your sample mean \( \bar{x} \).
• Determine the margin of error using your t-distribution critical value for \( \alpha/2 \) (often 0.025 for a 95% confidence) and multiply it by \( \frac{s}{\sqrt{n}} \).
• Calculate the confidence interval using: \( \bar{x} \pm t_{\alpha/2} \times \frac{s}{\sqrt{n}} \).

This interval provides a range within which we can be "confident" that the actual population mean resides. If our hypothesized mean falls outside of this interval, it adds weight against the null hypothesis. This level of confidence provides both reassurance and a functional understanding of any variability and potential errors in sampling.