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Chapter 10: Problem 82

c. A \(95 \%\) confidence interval for the difference between the mean level of tryptophan in the treated versus the control group is \((-0.007,0.004) .\) Interpret. Geneticists hypothesized that one reason for a prolonged lifespan observed in yeast cells treated with the common drug ibuprofen stems from inhibiting the uptake of certain amino acids. In an experiment, they compared levels of amino acids in cells treated with ibuprofen to cells from an untreated control group. For the amino acid tryptophan, the six measurements in the treated group had a mean level of 0.0067 with a standard deviation of 0.0038 . The six measurements in the control group had a mean level of 0.0080 with a standard deviation of 0.0050 . a. Identify the response variable and explanatory variable. Indicate whether each is quantitative or categorical. b. Find the standard error of the sample mean in each group and the standard error of the difference in the sample means.

Short Answer

Expert verified
The response variable is the mean level of tryptophan (quantitative), and the explanatory variable is treatment with ibuprofen (categorical). The standard errors are approximately 0.00155 for the treated group, 0.00204 for the control group, and 0.00255 for the difference in means.

Step by step solution

01

Identify the response and explanatory variables

In this experiment, we are interested in the effect of ibuprofen treatment on the level of tryptophan in yeast cells. The **response variable** is the mean level of tryptophan, which is **quantitative** as it represents a measurable quantity. The **explanatory variable** is whether the yeast cells were treated with ibuprofen or not, which is **categorical** as it classifies the cells into two groups: treated and control.
02

Calculate the standard error of the sample mean for the treated group

The formula for the standard error of the sample mean is given by: \( SE = \frac{s}{\sqrt{n}} \), where \( s \) is the standard deviation of the sample, and \( n \) is the sample size. For the treated group, we have \( s = 0.0038 \) and \( n = 6 \). Thus: \[ SE_{\text{treated}} = \frac{0.0038}{\sqrt{6}} \approx 0.00155 \]
03

Calculate the standard error of the sample mean for the control group

Similarly, for the control group, we use the same formula. The standard deviation \( s = 0.0050 \) and the sample size \( n = 6 \). Thus: \[ SE_{\text{control}} = \frac{0.0050}{\sqrt{6}} \approx 0.00204 \]
04

Calculate the standard error of the difference in sample means

The standard error for the difference in sample means is calculated using the formula: \[ SE_{\text{difference}} = \sqrt{SE_{\text{treated}}^2 + SE_{\text{control}}^2} \] where both \( SE_{\text{treated}} \) and \( SE_{\text{control}} \) are previously calculated. Thus: \[ SE_{\text{difference}} = \sqrt{0.00155^2 + 0.00204^2} \approx 0.00255 \]

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Key Concepts

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

Confidence Interval
A confidence interval is a range of values that likely contains the true difference between two population means. In this case, the experiment compares the mean tryptophan levels between the treated and control groups. When we say a 95% confidence interval, it implies there is a 95% chance that the true difference falls within the interval. For the measured difference in mean levels of tryptophan between the ibuprofen treated and control yeast cells, the confidence interval is given as (-0.007, 0.004).

This interval includes zero, indicating there is no statistically significant difference between the means at the 95% confidence level. It suggests that any observed differences could be due to chance rather than a real effect of the ibuprofen treatment.
  • If the interval included only negative numbers, it would suggest a lower tryptophan level in the treated group.
  • An interval with only positive numbers indicates the opposite.
Standard Error
The standard error provides a measure of how precise a sample mean is as an estimate of the true population mean. It is calculated as the standard deviation divided by the square root of the sample size. For the treated group with a standard deviation of 0.0038 and sample size of 6, the standard error is approximately 0.00155. For the control group, with a standard deviation of 0.0050 and sample size of 6, the standard error is roughly 0.00204.

Merging these, the standard error of the difference in sample means is calculated using both standard errors, which resulted in an approximate value of 0.00255.
  • Higher standard error values indicate more variability in the sample mean estimate.
  • Lower standard error values suggest greater precision and less variability.
This precision helps determine the reliability of the sample mean as an estimator for the population mean.
Response and Explanatory Variables
In any experiment, identifying the response and explanatory variables is crucial. Here, the **response variable** is the mean level of tryptophan, which is a quantitative measurement. It is what researchers are interested in understanding or predicting.

The **explanatory variable**, in this context, is the treatment of the yeast cells. It is categorical since it divides the yeast cells into two groups 鈥� treated with ibuprofen or untreated in the control group.
  • Response variables are typically numerical values we measure.
  • Explanatory variables are conditions or treatments applied to explain the changes.
Understanding these variables helps in structuring the analysis correctly and interpreting the results efficiently.
Quantitative and Categorical Variables
In statistics, variables can be classified as either quantitative or categorical. This classification importantly affects how data is analyzed.

Quantitative variables are numbers reflecting measurements or counts and allow operations like addition and comparison. In the experiment, the tryptophan level is a quantitative variable because it provides a measurable concentration.

Categorical variables describe characteristics or groups and are often labels like treated or untreated in this example. Such variables help in distinguishing among categories or classes without any numerical operations.
  • Quantitative variables help get precise measurements and infer relationships or differences between groups.
  • Categorical variables assist in classifying data, simplifying analysis by group comparisons.
Recognizing these variable types is fundamental in statistical analysis, guiding the appropriate methods and interpretations in data examination.

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Most popular questions from this chapter

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