A study of seat belt use involved children who were...

A study of seat belt use involved children who were hospitalized after motor vehicle crashes. For a group of 123 children who were wearing seat belts, the number of days in intensive care units (ICU) has a mean of $0.83$ and a standard deviation of $1.77$. For a group of 290 children who were not wearing seat belts, the number of days spent in ICUs has a mean of $1.39$ and a standard deviation of $3.06$ (based on data from "Morbidity Among Pediatric Motor Vehicle Crash Victims: The Effectiveness of Seat Belts," by Osberg and Di Scala, American Journal of Public Health, Vol. 82, No. 3$) .$ a. Use a $0.05$ significance level to test the claim that children wearing seat belts have a lower mean length of time in an ICU than the mean for children not wearing seat belts. b. Construct a confidence interval appropriate for the hypothesis test in part (a). c. What important conclusion do the results suggest?

A study of seat belt use involved children who were hospitalized after motor vehicle crashes. For a group of 123 children who were wearing seat belts, the number of days in intensive care units (ICU) has a mean of $0.83$ and a standard deviation of $1.77$. For a group of 290 children who were not wearing seat belts, the number of days spent in ICUs has a mean of $1.39$ and a standard deviation of $3.06$ (based on data from "Morbidity Among Pediatric Motor Vehicle Crash Victims: The Effectiveness of Seat Belts," by Osberg and Di Scala, American Journal of Public Health, Vol. 82, No. 3$) .$
a. Use a $0.05$ significance level to test the claim that children wearing seat belts have a lower mean length of time in an ICU than the mean for children not wearing seat belts.
b. Construct a confidence interval appropriate for the hypothesis test in part (a).
c. What important conclusion do the results suggest?

Key Concepts

Confidence Interval

A confidence interval provides a range of plausible values for the difference between population means and is constructed so that, under repeated sampling, a certain percentage (e.g., 95%) of such intervals would contain the true difference. When the confidence interval for the difference does not include zero, it suggests that there is a statistically significant difference between the groups, thereby supporting the hypothesis test’s conclusion.

Significance Level and p-Value

The significance level, often denoted as ? (alpha), represents the threshold probability for rejecting the null hypothesis, with ? = 0.05 in this instance. The p-value is the probability, computed from the test statistic, of observing a result at least as extreme as the one obtained if the null hypothesis were true. A p-value less than the chosen significance level indicates that the null hypothesis can be rejected.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis (H0) typically represents a statement of no effect or no difference. Conversely, the alternative hypothesis (Ha) represents the claim that is being tested—in this case, that the mean length of ICU stay is lower for children wearing seat belts compared to those not wearing seat belts.

Two-Sample t-Test

The two?sample t-test is used to compare the means from two independent groups when the population variances are unknown and possibly unequal. It assesses whether any observed difference between the sample means is statistically significant or could have arisen by random chance.

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence in a sample to support a claim about a population parameter. In this context, the testing procedure involves checking whether the mean number of ICU days for children wearing seat belts is significantly lower than that for children not wearing seat belts.

Transcript

00:01 So we will be assuming that the me number of days, for one, this is who were children who were wearing a seatbelt.

00:09 And this is for students or people who were not wearing a seatbelt.

00:15 And this is dealing with the number of days, that a mean number of days to stay in icu and the intensive care unit after an accident.

00:23 And so we're going to assume that they're equal.

00:26 The days are equal.

00:28 And alternately, that those wearing those.

00:30 A seatbelt will have a significantly lesser number of days in icu.

00:35 Now we are doing a two sample t test and our t value will be calculated by taking the difference between those means and the mean for those wearing was 0 .83 days minus the 1 .39 days.

00:58 Yeah it's only you know it's only 0 .5 days different.

01:01 Well, we have a standard deviation of the first divided by the number of samples, and that was a hundred twenty -three.

01:12 And the other standard deviation is 3 .06.

01:15 We are not assuming that their standard deviations are equal.

01:18 And this sample size was 290.

01:21 And this test statistic comes out to be a value of negative 2 .330.

01:33 And conservatively, we would use 122 for our degrees of freedom, but we're going to use the technology degrees of freedom or that formula.

01:43 I don't have my students memorize it, but the degrees of freedom of that is watch a lot bigger...