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Chapter 8: Problem 3

The mean 卤1 sd of In [calcium intake (mg)] among 25 females, 12 to 14 years of age, below the poverty level is \(6.56 \pm 0.64 .\) Similarly, the mean 卤1 sd of In [calcium intake (mg)] among 40 females, 12 to 14 years of age, above the poverty level is \(6.80 \pm 0.76\) What is the appropriate procedure to test for a significant difference in means between the two groups?

Short Answer

Expert verified
Use a two-sample t-test to determine if the difference in calcium intake means is significant.

Step by step solution

01

Understand the Problem

We need to determine if there is a significant difference in calcium intake between two groups of 12 to 14-year-old females: those below and above the poverty level. The means and standard deviations for each group are given.
02

Identify the Statistical Test

To compare the means of two independent groups, we use a two-sample t-test (also known as an independent t-test). This test will help determine if the difference in means is statistically significant.
03

Check Assumptions for T-Test

Ensure the data meets the assumptions for a t-test: the samples are independent, the data approximates a normal distribution (which is suggested by the reporting of means 卤 std deviations), and the variances are approximately equal. If variances are unequal, a Welch's t-test may be applied instead.
04

Perform the Independent T-Test

Use the sample sizes, means, and standard deviations provided to perform the two-sample t-test. The t-statistic can be calculated by the formula: \[ t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}} \]where \(\bar{x}_1\) and \(\bar{x}_2\) are the means, \(s_1\) and \(s_2\) are the standard deviations, and \(n_1\) and \(n_2\) are the sample sizes for each group.
05

Calculate Degrees of Freedom

The degrees of freedom (df) for the t-test can be estimated using: \[ df = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{\left(\frac{s_1^2}{n_1}\right)^2}{n_1 - 1} + \frac{\left(\frac{s_2^2}{n_2}\right)^2}{n_2 - 1}} \]
06

Determine the Critical Value

Using the calculated t-statistic and degrees of freedom, look up the critical value in a t-distribution table for the desired significance level (often 0.05 for a two-tailed test).
07

Conclude the Test

Compare the calculated t-statistic to the critical value. If the t-statistic is greater than the critical value, reject the null hypothesis and conclude the means are significantly different. Otherwise, do not reject the null hypothesis.

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Key Concepts

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

Understanding Calcium Intake Comparison
Comparing calcium intake between groups involves examining the average daily intake among different demographics. In this exercise, we focus on two groups of females aged 12 to 14: those living below poverty levels and those living above.
The calcium intake is reported as mean and standard deviation (sd) for each group.
This gives an idea about the average intake and the variability of intake within each group.

Understanding the averages is crucial as it helps identify any potentially critical differences that might exist due to socioeconomic factors.
When comparing two groups, we look for differences in their means to comprehensively understand how each group's average intake varies.
In research, these insights can lead to policy changes or intervention strategies aimed at closing the nutrition gap between these groups.
What is Statistical Significance Testing?
Statistical significance testing allows researchers to determine if observed differences in data are likely real or just occurred by chance.
In our case, the goal is to ascertain whether the difference in calcium intake between the two groups is statistically noteworthy.

A two-sample t-test is a method used here to assess that difference. It compares the means from two independent samples to see if they are significantly unlike provided their variability and sample size.
The null hypothesis in this context usually states there is no difference in means between the two groups.
If the computed t-statistic is greater than a critical value from the t-distribution table, we might conclude the difference is statistically significant 鈥� implying that it is unlikely due to random variance alone.
Key Assumptions for the T-Test
For the findings of a t-test to be valid, certain criteria must be met:
  • Independent Samples: Each sample must be independent of the other, meaning the subjects in one group do not affect those in the other.
  • Normal Distribution: The data within each group should approximate a normal distribution. If the sample sizes are large enough, the t-test can be robust even if the data departs slightly from normality.
  • Equal Variances: The variances of the two groups should be similar. If there's a significant discrepancy, a Welch's t-test, which does not assume equal variances, should be considered.
Checking these assumptions is crucial, as violating them could lead to incorrect conclusions.
If everything checks out, the two-sample t-test provides a reliable method to evaluate the calcium intake differences between the two groups under study.

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Most popular questions from this chapter

A possible important environmental determinant of lung function in children is the amount of cigarette smoking in the home. Suppose this question is studied by selecting two groups: Group 1 consists of 23 nonsmoking children 5-9 years of age, both of whose parents smoke, who have a mean forced expiratory volume (FEV) of 2.1 L and a standard deviation of \(0.7 \mathrm{L} ;\) group 2 consists of 20 nonsmoking children of comparable age, neither of whose parents smoke, who have a mean FEV of \(2.3 \mathrm{L}\) and a standard deviation of \(0.4 \mathrm{L}\). Provide a 95\% Cl for the true mean difference in FEV between \(5-\) to 9 -year- old children whose parents smoke and comparable children whose parents do not smoke.

A possible important environmental determinant of lung function in children is the amount of cigarette smoking in the home. Suppose this question is studied by selecting two groups: Group 1 consists of 23 nonsmoking children 5-9 years of age, both of whose parents smoke, who have a mean forced expiratory volume (FEV) of 2.1 L and a standard deviation of \(0.7 \mathrm{L} ;\) group 2 consists of 20 nonsmoking children of comparable age, neither of whose parents smoke, who have a mean FEV of \(2.3 \mathrm{L}\) and a standard deviation of \(0.4 \mathrm{L}\). What are the appropriate null and alternative hypotheses to compare the means of the two groups?

Insulin-like growth factor 1 (IGF- 1 ) is a hormone that plays an important role in childhood growth and may be associated with several types of cancer. In some studies, it is measured in serum, and in other studies it is measured in plasma. Blood plasma is prepared by spinning a tube of fresh blood containing an anti-coagulant in a centrifuge until the blood cells fall to the bottom. The blood plasma is then poured or drawn off. Blood serum is blood plasma without clotting factors (i.e., whole blood minus both the cells and the clotting factors). In a multi-center study, IGF-1 was measured in some centers with serum and in other centers with plasma. The following results were obtained: Assuming that the distribution of IGF-1 is approximately normal, what test can be used to compare the mean IGF-1 obtained from the two sources?

A study compared mean electroretinogram (ERG) amplitude of patients with different genetic types of retinitis pigmentosa (RP), a genetic eye disease that often results in blindness. The results shown in Table 8.24 were obtained for In (ERG amplitude) among patients \(18-29\) years of age. $$\begin{array}{lcc} \hline \text { Genetic type } & \text { Mean } \pm \mathrm{sd} & n \\ \hline \text { Dominant } & 0.85 \pm 0.18 & 62 \\ \text { Recessive } & 0.38 \pm 0.21 & 35 \\ \text { X-linked } & -0.09 \pm 0.21 & 28 \\ \hline \end{array}$$ What is the standard error of In(ERG amplitude) among patients with dominant RP? How does it differ from the standard deviation in the table?

The mean 卤1 sd of In [calcium intake (mg)] among 25 females, 12 to 14 years of age, below the poverty level is \(6.56 \pm 0.64 .\) Similarly, the mean 卤1 sd of In [calcium intake (mg)] among 40 females, 12 to 14 years of age, above the poverty level is \(6.80 \pm 0.76\) Test for a significant difference between the variances of the two groups.
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