# Is statistical inference intuitive to babies? In other words, are babies able to generalize from sample to population?

Question
Comparing two groups

Is statistical inference intuitive to babies? In other words, are babies able to generalize from sample to population? In this study,1 8-month-old infants watched someone draw a sample of five balls from an opaque box. Each sample consisted of four balls of one color (red or white) and one ball of the other color. After observing the sample, the side of the box was lifted so the infants could see all of the balls inside (the population). Some boxes had an “expected” population, with balls in the same color proportions as the sample, while other boxes had an “unexpected” population, with balls in the opposite color proportion from the sample. Babies looked at the unexpected populations for an average of 9.9 seconds ($$sd = 4.5$$ seconds) and the expected populations for an average of 7.5 seconds ($$sd = 4.2$$ seconds). The sample size in each group was 20, and you may assume the data in each group are reasonably normally distributed. Is this convincing evidence that babies look longer at the unexpected population, suggesting that they make inferences about the population from the sample? Let group 1 and group 2 be the time spent looking at the unexpected and expected populations, respectively. A) Calculate the relevant sample statistic. Enter the exact answer. Sample statistic: _____

B) Calculate the t-statistic. Round your answer to two decimal places. t-statistic = ___________

C) Find the p-value. Round your answer to three decimal places. p-value =__________

2020-12-29

From the given information , for group 1(unexpected): $$\displaystyle\overline{{{x}}}_{{1}}={9.9},{s}_{{1}}={4.5},{n}={20},$$ for group 2(expected): $$\displaystyle\overline{{{x}}}_{{2}}={7.5},{s}_{{2}}={4.2},$$ and $$\displaystyle{n}_{{2}}={20}.$$ Null hypothesis $$\displaystyle{H}_{{0}}:\mu_{{1}}=\mu_{{2}}$$ Alternative hypothesis $$\displaystyle{H}_{{\alpha}}:\mu_{{1}}{>}\mu_{{2}}$$

(A) Sample statistic $$\displaystyle\mu_{{{\left({x}_{{1}}-{x}_{{2}}\right)}}}=\overline{{{x}}}_{{1}}-\overline{{{x}}}_{{2}}={9.9}-{7.5}={2.4}$$

(B) The test statistic $$\displaystyle{t}_{{{c}\alpha{1}}}{\frac{{\overline{{{x}}}_{{1}}-\overline{{{x}}}_{{2}}}}{{\sqrt{{{\frac{{{{S}_{{{1}}}^{{{2}}}}}}{{{n}_{{1}}}}}+{\frac{{{{S}_{{{2}}}^{{{2}}}}}}{{{n}_{{2}}}}}}}}}}={\frac{{{9.9}-{7.5}}}{{{\frac{{{\left({4.5}\right)}^{{2}}}}{{{20}}}}+{\frac{{{\left({4.2}\right)}^{{2}}}}{{{20}}}}}}}={1.7437}$$

Degrees of freedom $$\displaystyle={\left({n}_{{1}}+{n}_{{2}}-{2}\right)}={38}$$

(C) $$\displaystyle{P}-{v}{a}{l}{u}{e}={P}{\left({t}{>}{1.7437}\right)}={0.045},$$

$${P}-{v}{a}{l}{u}{e}={0.045}$$ less than the level of significance $$\displaystyle\alpha={0.05},$$ we reject the null hypothesis.

### Relevant Questions

A two-sample inference deals with dependent and independent inferences. In a two-sample hypothesis testing problem, underlying parameters of two different populations are compared. In a longitudinal (or follow-up) study, the same group of people is followed over time. Two samples are said to be paired when each data point in the first sample is matched and related to a unique data point in the second sample.
This problem demonstrates inference from two dependent (follow-up) samples using the data from the hypothetical study of new cases of tuberculosis (TB) before and after the vaccination was done in several geographical areas in a country in sub-Saharan Africa. Conclusion about the null hypothesis is to note the difference between samples.
The problem that demonstrates inference from two dependent samples uses hypothetical data from the TB vaccinations and the number of new cases before and after vaccination. $$\begin{array}{|c|c|} \hline Geographical\ regions & Before\ vaccination & After\ vaccination\\ \hline 1 & 85 & 11\\ \hline 2 & 77 & 5\\ \hline 3 & 110 & 14\\ \hline 4 & 65 & 12\\ \hline 5 & 81 & 10\\\hline 6 & 70 & 7\\ \hline 7 & 74 & 8\\ \hline 8 & 84 & 11\\ \hline 9 & 90 & 9\\ \hline 10 & 95 & 8\\ \hline \end{array}$$
Using the Minitab statistical analysis program to enter the data and perform the analysis, complete the following: Construct a one-sided $$\displaystyle{95}\%$$ confidence interval for the true difference in population means. Test the null hypothesis that the population means are identical at the 0.05 level of significance.

The sampling distribution of $$\frac{S_{1}^{2}}{S_{2}^{2}}$$ is the $$\chi^2$$ distribution. Give a correct answer true or false inference for two population variances is done through their difference.

Iron is very important for babies' growth. A common belief is that breastfeeding will help the baby to get more iron than formula feeding. To justify the belief, a study followed 2 groups of babies from born to 6 months. With one group babies are breast fed, and the other group are formula fed without iron supplements. Data below shows iron levels of those two groups of babies.

$$\begin{array}{|c|c|} \hline Group & Sample\ size & mean & Standard\ deviation \\ \hline Breast-fed & 23 & 13.3 & 1.7 \\ \hline Formula-fed & 23 & 12.4 & 1.8 \\ \hline DIFF = Breast-Formula & 23 & 0.9 & 1.4 \\ \hline \end{array}$$

(1) There are two groups we need to compare for the study: Breast-Fed and Formula- Fed. Are those two groups dependent or independent? Based on your answer, what inference procedure should we apply for this research?

(2) Please perform the inference you decided in (1), and make sure to follow the 5-step procedure for any hypothesis test.

(3) Based on your conclusion in (2), what kind of error could you make? Explain the type of error using the context words for this research

As a vaccine scientist, you are required to test your newly developed vaccine in two different populations, populations Xand Y to ensure the safety and effectiveness of the vaccine. There are 3190 subjects from database X and 6094 subjects from database Therefore, you must select a number of subjects from populations X and Y to form a group. The newly formed of group must consist of subjects from both populations without repetition. The maximum number of groups which can be formed is denoted as d.
(1) Use Euclidean algorithm to find $$d= GCD(X, Y).$$
2) Find the integers s and tsuch that $$d = sX + tY$$
3) With the answer obtained from a, what is the ratio of subjects selected from population $$X\ and\ Y, PX : PY.$$
4) Find Least Common Multiple for $$X\ and\ Y, LCM(X, Y).$$

$$\begin{array}{|c|c|} \hline & Housework Hours \\ \hline Gender & Sample\ Size & Mean & Standard\ Deviation \\ \hline Women & 473473 & 33.133.1 & 14.214.2 \\ \hline Men & 488488 & 18.618.6 & 15.715.7 \\ \end{array}$$

a. Based on this​ study, calculate how many more hours per​ week, on the​ average, women spend on housework than men.

b. Find the standard error for comparing the means. What factor causes the standard error to be small compared to the sample standard deviations for the two​ groups? The cause the standard error to be small compared to the sample standard deviations for the two groups.

c. Calculate the​ 95% confidence interval comparing the population means for women Interpret the result including the relevance of 0 being within the interval or not. The​ 95% confidence interval for ​$$\displaystyle{\left(\mu_{{W}}-\mu_{{M}}​\right)}$$ is: (Round to two decimal places as​ needed.) The values in the​ 95% confidence interval are less than 0, are greater than 0, include 0, which implies that the population mean for women could be the same as is less than is greater than the population mean for men.

d. State the assumptions upon which the interval in part c is based. Upon which assumptions below is the interval​ based? Select all that apply.

A.The standard deviations of the two populations are approximately equal.

B.The population distribution for each group is approximately normal.

C.The samples from the two groups are independent.

D.The samples from the two groups are random.

Situations comparing two proportions are described. In each case, determine whether the situation involves comparing proportions for two groups or comparing two proportions from the same group (1 point each) a. Compare the proportion of U.S. adults who have a positive opinion about the media versus those who have a negative opinion about the media. A) Comparing proportions for two groups B) Comparing two proportions from the same group b. Compare the proportion of female students at a university who live in a dorm to the proportion of male students at a university who live in a dorm A) Comparing proportions for two groups B) Comparing two proportions from the same group

Give a full answer to given question

1) An independent-measures study uses ?

2) An independent-measures study uses two samples, each with $$n = 13$$, to compare two treatment conditions. What is the df value for the t statistic for this study?

To explain: why the combination $$\binom{n}{r}=\binom{n}{r,n-r}$$