Child care hours, Part I. The China Health and Nutrition...

Child care hours, Part I. The China Health and Nutrition Survey aims to examine the effects of the health, nutrition, and family planning policies and programs implemented by national and local governments. One of the variables collected on the survey is the number of hours parents spend taking care of children in their household under age 6 (feeding, bathing, dressing, holding, or watching them). In 2006,487 females and 312 males were surveyed for this question. On average, females reported spending 31 hours with a standard deviation of 31 hours, and males reported spending 16 hours with a standard deviation of 21 hours. Calculate a $95 \%$ confidence interval for the difference between the average number of hours Chinese males and females spend taking care of their children under age 6 . Also comment on whether this interval suggests a significant difference between the two population parameters. You may assume that conditions for inference are satisfied. $^{34}$

Key Concepts

Assumptions for Inference

Inference methods such as constructing a confidence interval rely on certain assumptions, including the independence of observations, the appropriateness of using a t-distribution (normally when sample sizes are moderate and population variances are unknown), and the assumption that the data are approximately normally distributed. Meeting these conditions ensures the validity of the interval estimate and related statistical tests.

Statistical Significance and Hypothesis Testing

Statistical significance in this context refers to determining whether the observed difference between group means is likely to be present in the population or could have occurred by chance. If the confidence interval for the difference between the means does not include zero, it suggests that the difference is statistically significant at the chosen confidence level, indicating a real difference between the population parameters.

Standard Error Calculation

The standard error is a measure of the variability in the sampling distribution of a statistic, such as the difference between two means. It is calculated using the sample standard deviations and sample sizes, and it quantifies the uncertainty associated with the sample estimates. This concept is essential for constructing confidence intervals and conducting hypothesis tests.

Confidence Interval for Difference of Means

A confidence interval for the difference between two means provides a range of values within which the true difference in population means is expected to lie, with a given level of confidence (e.g., 95%). This interval is particularly useful for understanding the magnitude and direction of differences between groups and for determining whether an observed difference might be due to sampling variability.

Transcript

00:01 In china, four -year -olds average three hours a day unsupervised.

00:06 Most of the unsupervised children live in rural areas considered safe.

00:10 Suppose that the standard deviation is 1 .5 hours and the amount of time spent alone is normally distributed.

00:18 We randomly select one chinese four -year -old living in a rural area.

00:23 We are interested in the amount of time the child spends alone per day.

00:31 In words, define the random variable x.

00:35 Well, because x is going to be what we are interested in, and we are interested in the amount of time a child spends alone, we can put that into words.

00:45 So we're going to say that x is the number of hours that a chinese four -year -old spends unsupervised during the day.

00:54 B, x can be defined by a normal distribution or a mean of three in a standard deviation at 1 .5 hours.

01:08 C.

01:10 Find their probability that a child spends less than one hour per day unsupervised.

01:16 So x being less than one.

01:23 So the picture can help us see what's going on here in which we have a mean of three and a standard deviation of 1 .5.

01:33 We'll just draw a few standard deviations on each side.

01:42 We want to know what's the probability that they spend less than one hour.

01:51 So for this problem, we can use our normal cdf function.

01:56 In our calculator.

01:59 You can insert your lower bound to be negative affinity or you can insert it as zero because it's impossible to be insupervised less than zero hours in a day...

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