In his book Outliers The Story of Success Malcolm Gladwell a

In his book Outliers: The Story of Success, Malcolm Gladwell argues that Chinese children have a built-in advantage over American children when it comes to learning mathematics because Chinese words for numbers have fewer syllables than English words for numbers. Gladwell cites research that Chinese children can count higher at earlier ages than American children can, specifically mentioning that 4-year-old Chinese children can count to a higher number, on average, compared to 4-year-old American children. Suppose that you want to conduct a study to try to replicate this result.

a) Identify the explanatory and response variables in this study. Ideally, would your study make use of random sampling, random assignment, or both? Explain.

b) Identify (in words and symbols) the parameter(s) that you would be interested in. State the appropriate hypotheses for testing whether Chinese 4-year-olds really can count to a higher number, on average, compared to American 4-year-olds.

c) Suppose you find that Chinese children can count to 40 on average, compared to 15 for American children. What further information would you need to conduct a two-sample t-test? Would this difference (between 40 and 15) be more, less, or equally significant if the sample sizes were 50 in each group or 250 in each group?

d) Would this difference (between 40 and 15) be more, less, or equally significant if the individual students’ values (for how high they can count) were very spread out or not very spread out?

Solution

a) We are interested in comparing the counting ability of Chinese children aged 4 years with American children of the same age.

Explanatory variables or independent variables will be the scores of both the children and

response variable here would be the difference between the scores, std error and test statistic.

The study will make of both random sampling and random assignment to ensure accurate results.

b) Average scores of both the children, sample size and std deviation of the sample

H0: mu_c = mu_a

Ha: mu_c>mu_a

(Right tailed test for comparing muc average of chinese children scores with mua average of American children\'s)

c) Further information we require is the std deviation of each sample, and whether randomness is assured.

We cannot find whether difference will be more or less actually without knowing std deviation but we can say that for sample sizes large, maximum accuracy of the result can be expected.

d) If the individual students were very spread out then std deviation would be very high.

Hence std error will be high and hence test statistic will be low.

If test statistic is low chances of rejection of null hypohtesis is more or difference would be more.

If not very spread out mean difference would be less.

c)