In a study aimed at reducing developmental problems in...

In a study aimed at reducing developmental problems in lowbirth-weight babies (under 2500 grams), 347 infants were exposed to a special educational curriculum while 561 did not receive any special help. After 3 years, the children exposed to the special curriculum showed a mean IQ of 93.5 with a standard deviation of 19.1; the other children had a mean IQ of 84.5 with a standard deviation of $19.9 .$ Find a $95 \%$ confidence interval estimate for the difference in mean IQs of all low-birth-weight babies who receive special intervention and those who do not. (A) $(93.5-84.5) \pm 1.97 \sqrt{\frac{(19.1)^{2}}{347}+\frac{(19.9)^{2}}{561}}$ (B) $(93.5-84.5) \pm 1.97\left(\frac{19.1}{\sqrt{307}}+\frac{19.9}{\sqrt{561}}\right)$ (C) $(93.5-84.5) \pm 1.97 \sqrt{\frac{(19.1)^{2}}{347}+\frac{(19.9)^{2}}{561}}$ (D) $(93.5-84.5) \pm 1.65\left(\frac{19.1}{\sqrt{37}}+\frac{19.9}{\sqrt{561}}\right)$ (E) $(93.5-84.5) \pm 1.65 \sqrt{\frac{(19.1)^{2}+(19.9)^{2}}{347+561}}$

In a study aimed at reducing developmental problems in lowbirth-weight babies (under 2500 grams), 347 infants were exposed to a special educational curriculum while 561 did not receive any special help. After 3 years, the children exposed
to the special curriculum showed a mean IQ of 93.5 with a standard deviation of 19.1; the other children had a mean IQ
of 84.5 with a standard deviation of $19.9 .$ Find a $95 \%$ confidence interval estimate for the difference in mean IQs of all low-birth-weight babies who receive special intervention and those who do not.
(A) $(93.5-84.5) \pm 1.97 \sqrt{\frac{(19.1)^{2}}{347}+\frac{(19.9)^{2}}{561}}$
(B) $(93.5-84.5) \pm 1.97\left(\frac{19.1}{\sqrt{307}}+\frac{19.9}{\sqrt{561}}\right)$
(C) $(93.5-84.5) \pm 1.97 \sqrt{\frac{(19.1)^{2}}{347}+\frac{(19.9)^{2}}{561}}$
(D) $(93.5-84.5) \pm 1.65\left(\frac{19.1}{\sqrt{37}}+\frac{19.9}{\sqrt{561}}\right)$
(E) $(93.5-84.5) \pm 1.65 \sqrt{\frac{(19.1)^{2}+(19.9)^{2}}{347+561}}$

Key Concepts

Confidence Interval for the Difference in Means

A confidence interval for the difference in means estimates the range of plausible values for the difference between two population means based on sample data. It is built around the sample difference and incorporates a margin of error calculated from the standard error and a critical value corresponding to the desired level of confidence.

Standard Error of the Difference

This concept involves quantifying the variability in the estimated difference between two sample means. The standard error is calculated as the square root of the sum of the variances from each group (each divided by its respective sample size). It represents how much the difference in sample means is expected to vary from sample to sample.

Critical Value

The critical value is a multiplier derived from a probability distribution (such as the t or normal distribution) that corresponds to the desired level of confidence (for example, 95%). This value is used to scale the standard error in order to define the margin of error for the confidence interval.

Two-Sample Comparison

This concept refers to the statistical approach used when comparing the means of two independent groups. It involves using formulas that combine information from both groups—such as their means, standard deviations, and sample sizes—to assess whether the observed difference is statistically significant.

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