$4$ . Baby walkers are seats hanging from frames that allow babies to sit upright with their legs dangling and feet touching the floor. Walkers have wheels on their legs that allow the infant to propel the walker around the house long before he or she can walk or even crawl. Typically, babies use walkers between the ages of $4$ months and $11$ months. $\newline$ Because most walkers have tray tables in front that block babies' views of their feet, child psychologists have begun to question whether walkers affect infants' cognitive development. One study compared mental skills of a random sample of those who used walkers with a random sample of those who never used walkers. Mental skill scores averaged $113$ for $54$ babies who used walkers (standard deviation of $12$ ) and $123$ for $55$ babies who did not use walkers (standard deviation of $15$ ). $\newline$ (a) Is there evidence that the mean mental skill score of babies who use walkers is different from the mean mental skill score of babies who do not use walkers? Explain your answer.

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Algebra 2

Interpret confidence intervals for population means

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. Baby walkers are seats hanging from frames that allow babies to sit upright with their legs dangling and feet touching the floor. Walkers have wheels on their legs that allow the infant to propel the walker around the house long before he or she can walk or even crawl. Typically, babies use walkers between the ages of

4

months and

11

months.

\newline

Because most walkers have tray tables in front that block babies' views of their feet, child psychologists have begun to question whether walkers affect infants' cognitive development. One study compared mental skills of a random sample of those who used walkers with a random sample of those who never used walkers. Mental skill scores averaged

113

for

54

babies who used walkers (standard deviation of

12

) and

123

for

55

babies who did not use walkers (standard deviation of

15

\newline

(a) Is there evidence that the mean mental skill score of babies who use walkers is different from the mean mental skill score of babies who do not use walkers? Explain your answer.

Calculate Mean Difference: Calculate the mean difference between the two groups. $\newline$ Mean difference $= \text{Mean of non-walker group} - \text{Mean of walker group}$ $\newline$ $= 123 - 113$ $\newline$ $= 10$
Calculate Standard Error: Calculate the standard error of the difference between two independent means. $\newline$ $SE = \sqrt{(SD1^2 / n1) + (SD2^2 / n2)}$ $\newline$ $= \sqrt{(12^2 / 54) + (15^2 / 55)}$ $\newline$ $= \sqrt{(144 / 54) + (225 / 55)}$ $\newline$ $= \sqrt{2.6667 + 4.0909}$ $\newline$ $= \sqrt{6.7576}$ $\newline$ $= 2.6$
Calculate t-score: Calculate the t-score for the difference between the two means. $\newline$ $t = \frac{\text{Mean difference}}{\text{SE}}$ $\newline$ $= \frac{10}{2.6}$ $\newline$ $= 3.85$
Determine Degrees of Freedom: Determine the degrees of freedom for the t-test. $\newline$ $df = \min(n_1 - 1, n_2 - 1)$ $\newline$ $= \min(54 - 1, 55 - 1)$ $\newline$ $= \min(53, 54)$ $\newline$ $= 53$
Look up Critical t-value: Look up the critical t-value for a two-tailed test at a significance level of $0.05$ with $53$ degrees of freedom. $\newline$ Assuming the critical t-value is approximately $2.00$ (this would be checked in a t-distribution table or using statistical software).
Compare t-score to Critical t-value: Compare the calculated t-score to the critical t-value. Since $3.85 > 2.00$ , the calculated t-score is in the rejection area.
Conclude Significant Difference: Conclude whether there is a significant difference between the two means. Since the calculated $t$ -score is greater than the critical $t$ -value, we reject the null hypothesis that there is no difference between the mean mental skill scores of babies who use walkers and those who do not.