A study was conducted to assess the association between...

A study was conducted to assess the association between climate conditions in infancy and adult blood pressure and anthropometric measures (e.g., height, weight) [19]. There were 3964 British women born between 1919 and 1940 who were divided into quartiles ( $n=991$ per quartile) according to mean summer temperature $\left(^{\circ} \mathrm{C}\right)$ in the first year of life. The data in Table 8.34 were presented. We will assume that the distribution of adult height within a quartile is normally distributed and that the sample sizes are large enough that the $t$ distribution can be approximated by a normal distribution. 8.143 What test can be performed to compare the mean adult height between the first (Q1) and the fourth (Q4) quartiles? (Assume that the underlying variances of adult height in $\mathrm{Q} 1$ and $\mathrm{Q} 4$ are the same.)

A study was conducted to assess the association between climate conditions in infancy and adult blood pressure and anthropometric measures (e.g., height, weight) [19]. There were 3964 British women born between 1919 and 1940 who were divided into quartiles ( $n=991$ per quartile) according to mean summer temperature $\left(^{\circ} \mathrm{C}\right)$ in the first year of life. The data in Table 8.34 were presented. We will assume that the distribution of adult height within a quartile is normally distributed and that the sample sizes are large enough that the $t$ distribution can be approximated by a normal distribution.
8.143 What test can be performed to compare the mean adult height between the first (Q1) and the fourth (Q4) quartiles? (Assume that the underlying variances of adult height in $\mathrm{Q} 1$ and $\mathrm{Q} 4$ are the same.)

Key Concepts

Independent Two-Sample t-Test

This is a statistical method used to compare the means of two independent groups. In the context of the question, it is applied to assess whether the mean adult height between the first and fourth quartiles differs significantly. The test computes a t-statistic by considering the difference between group means relative to the variability observed within the groups.

Equal Variance Assumption

The assumption of equal variances (homoscedasticity) implies that the variability in the measurements (adult height) is consistent across the two groups being compared. When this condition is met, a pooled estimate of the variance is used in the t-test, which simplifies the analysis and improves the reliability of the test statistic.

Normal Distribution Assumption

Assuming that the distribution of adult height within each quartile is normally distributed is fundamental for the validity of the t-test. Normality ensures that the sampling distribution of the test statistic approximates a t-distribution, allowing for accurate inference about the population means.

Large Sample Size and Normal Approximation

With large sample sizes, the t-distribution approximates the normal distribution due to the central limit theorem. This approximation justifies the use of the t-test even when the underlying population distribution may not be perfectly normal, as the distribution of the sample means tends toward normality with increased sample size.

Transcript

00:01 The following is a solution to number nine, and this is a study to see if this is a normal distribution.

00:06 It's a number of days.

00:08 There's a 20 -year study number of days in july for, i think it's like kit carson, colorado, and see if those temperatures, the average daily temperature, see if it's approximately normal.

00:18 And there's two parts to this.

00:19 The first part, i actually just kind of typed out because it'd be easier this way.

00:22 But the way that they structure the percentages, that represents the empirical rule.

00:26 So if this actually matches a normal distribution, that means it should basically, follow the empirical rule, which means 68 % of the data lies within one standard deviation in the mean.

00:36 So that's why they had 34 and 34 % that were minus one standard deviation plus one standard deviation from the mean.

00:44 That's why it was between 67 and 83.

00:47 In fact, i'll write that down.

00:48 So 68 % of data between 67 degrees and 83 degrees.

01:03 Okay, so this is because if you take 65, or sorry, 75 minus 8, you get 67, and then 75 plus 8 you get the 83.

01:11 And then likewise, 95 % of the data lies within two standard deviations in the means.

01:16 So that's where we get, kind of think we're starting in the middle of the hump, and then we just kind of work back and work out that way.

01:22 And 13 .5 plus 34 plus 34 plus 13 .5, that equals 95%.

01:26 So 95 % of the data is between 59 degrees and 91 degrees.

01:37 And then finally, basically every data value, at least it should be, otherwise it's probably an outlier, but anything else, about 99 .7 % of the data, which is the rest of it.

01:48 So 2 .35 plus 13 .5 plus 34, and then back down the hill, that equals 99 .7.

01:54 So 99 .7 % of the data is within three standard deviations of the mean.

02:00 And that three standard deviations, if you just subtract 8 and add 8, you get 51 degrees.

02:06 So basically all these data values should be between 51 and 99.

02:10 So that's the empirical rule.

02:12 That's the first part.

02:14 Okay, so the second part is just like normal.

02:16 So we have these observed values i was given to.

02:18 And then the expected, like normal, i just took the percent times the sample size and i got these expected.

02:25 And what's nice about this is you just really just have to do half of it because the other half should mirror image.

02:30 So that's where i get those numbers.

02:32 So let's go and answer these questions.

02:33 The first part it says what's the significance level, and that's the alpha, and that's given to you as 1%.

02:40 Okay, and then we do the null and the alternative, and again, you can probably write this however you want, but basically the null is going to say that these daily temperatures is approximately normal, and then the alternative is that they're not normal.

02:55 So in prettier words, let's go ahead and say the average daily temperature.

03:01 I'm going to abbreviate here, just because of room.

03:05 In july follows, and this is kit carson, colorado follows a normal distribution with mean 75 and standard deviation 8.

03:24 And i should probably put units there degrees...

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