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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. $$ \begin{array}{rrrrr} 421.0 & 452.6 & 456.1 & 494.6 & 373.8 \\ 90.5 & 110.7 & 96.4 & 81.7 & 102.4 \\ 241.0 & 296.0 & 317.0 & 290.9 & 256.5 \\ 447.8 & 687.6 & 705.7 & 879.0 & 88.8 \\ 296.0 & 273.0 & 268.0 & 227.5 & 279.3 \\ 258.5 & 296.0 & & & \end{array} $$ 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

State the Hypotheses

We are testing the hypothesis that the mean body weight of guinea pigs at birth is 300 grams. Define the null hypothesis as \( H_0: \mu = 300 \) and the alternative hypothesis as \( H_a: \mu eq 300 \). This is a two-tailed test.

02

Gather Data and Calculate Mean

Collect and list all the body weight data points available: 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. Calculate the sample mean \( \bar{x} \).

03

Calculate Standard Deviation

Find the standard deviation of the data set. Calculate the variance by finding the average of the squared differences from the mean, then take the square root to find the standard deviation.

04

Conduct the t-test

Use the t-test formula: \( t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}} \), where \( \bar{x} \) is the sample mean, \( \mu \) is the hypothesized population mean (300 grams), \( s \) is the standard deviation, and \( n \) is the sample size. Calculate the t-value.

05

Determine Critical Value and Decision

Determine the critical t-value for a two-tailed test with \( n-1 \) degrees of freedom at \( \alpha = 0.05 \). Compare the calculated t-value to this critical value to decide whether to reject or fail to reject the null hypothesis.

06

Calculate P-Value

Find the p-value corresponding to the calculated t-value. This is the smallest level of significance at which the null hypothesis can be rejected. Use a t-distribution table or software for this calculation.

07

Confidence Interval Approach

Instead of the hypothesis test, calculate a two-sided confidence interval for the mean. The formula is \( \bar{x} \pm t_{critical} \times \frac{s}{\sqrt{n}} \). Check if 300 lies within this range to draw conclusions similar to the hypothesis test.

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Key Concepts

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

Two-tailed Test

A two-tailed test is a statistical method used to determine if a sample mean significantly differs from a known or hypothesized population mean. This type of test is useful when you expect the value could be either lower or higher than the expected value, hence the term "two-tailed." In the guinea pig study, we want to check if the mean body weight at birth is exactly 300 grams or not, which makes this a perfect example of a two-tailed test.

• Null Hypothesis (\(H_0\)): The mean weight is 300 grams.
• Alternative Hypothesis (\(H_a\)): The mean weight is not 300 grams (could be either more or less).

When conducting a two-tailed test, the critical region for rejection of the null hypothesis lies in both tails of the normal distribution curve. This means you will compare your test statistic to both the upper and lower critical values, increasing the chances of testing for any significant difference in either direction. Using a two-tailed test ensures that either an increase or a decrease in the parameter being tested is detected.

Confidence Interval

A confidence interval gives a range of values, calculated from the sample data, which is likely to contain the population parameter (mean in this case) with a specified level of confidence. In the guinea pig study, we can use a two-sided confidence interval to assess if the mean body weight is 300 grams.

• A 95% confidence interval gives a 95% certainty that the true mean falls within this range.
• The formula for the confidence interval is \(\bar{x} \pm t_{critical} \times \frac{s}{\sqrt{n}}\), where \(\bar{x}\) is the sample mean, \(s\) is the standard deviation, and \(n\) is the sample size, \(t_{critical}\) is the t value for the desired level of confidence.
• If 300 grams falls within this interval, we do not have sufficient evidence to reject the null hypothesis at the given confidence level.

A confidence interval provides more informative results than a simple hypothesis test result because it presents the range in which the true parameter likely lies, offering more insight into the data's implications.

t-test

The t-test is a fundamental statistical test used to compare the sample mean to the population mean when the sample size is small, and the population standard deviation is unknown. It's particularly useful when the sample size is insufficient to rely on the normal distribution formula. In the case of the guinea pig weights, we utilize the t-test due to the relatively small sample size.

• The formula for the t-test is \( t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}} \), where \(\bar{x}\) is the sample mean, \(\mu\) is the hypothesized mean (300 grams), \(s\) is the sample standard deviation, and \(n\) is the sample size.
• This formula calculates the t-value, which measures how many standard deviations the sample mean is from the hypothesized mean.
• After finding the t-value, compare it with the critical t-value from the t-distribution table to decide whether to reject the null hypothesis.

The t-test essentially helps determine if there is a statistically significant difference between the sample mean and population mean, providing a foundation for decision-making.

Level of Significance

The level of significance is a threshold value that determines the probability of making a Type I error, which is rejecting the null hypothesis when it is actually true. It is denoted by \(\alpha\) and represents the risk one is prepared to take when concluding that a change or difference exists.

• Common levels of significance are 0.05, 0.01, and 0.10, with the most widely used being 0.05 (5%), which indicates a 5% chance of incorrectly rejecting the null hypothesis.
• In the guinea pig weight study, \(\alpha\) is set to 0.05.
• If the p-value is less than \(\alpha\), the results are considered statistically significant, meaning we would reject the null hypothesis.

Understanding the level of significance is crucial because it defines the reliability of the test results. By choosing a level of significance, researchers decide how much evidence against the null hypothesis they require before accepting that results are not due to random chance.