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A big-toe problem Hallux abducto valgus (call it HAV) is a deformation of the big toe that is fairly uncommon in youth and often requires surgery. Doctors used X-rays to measure the angle (in degrees) of deformity in a random sample of patients under the age of 21 who came to a medical center for surgery to correct HAV. The angle is a measure of the seriousness of the deformity. For these 21 patients, the mean HAV angle was 24.76 degrees and the standard deviation was 6.34 degrees. A dotplot of the data revealed no outliers or strong skewness. \(^{27}\) (a) Construct and interpret a 90% confidence interval for the mean HAV angle in the population of all such patients. (b) Researchers omitted one patient with an HAV angle of 50 degrees from the analysis due to a measurement issue. What effect would including this outlier have on the confidence interval in (a)? Justify your answer.

Step by step solution

01

Identify the Problem Type

This problem involves constructing a confidence interval for the mean of a population based on a sample. The objective is to estimate the mean HAV angle from the sample of patients.

02

Identify Given Information

The sample mean is 24.76 degrees, the sample standard deviation is 6.34 degrees, and the sample size is 21. The sample is assumed to be drawn from a normally distributed population or is sufficiently large for the Central Limit Theorem to apply.

03

Determine the Critical Value

For a 90% confidence interval for the mean, we need to find the t critical value using a t-distribution table. With a sample size of 21, the degrees of freedom is 20. Look up the t* value for 90% confidence: t* 鈮� 1.725.

04

Calculate the Standard Error of the Mean (SEM)

The SEM is the standard deviation divided by the square root of the sample size: \[ \text{SEM} = \frac{6.34}{\sqrt{21}} \approx 1.38. \]

05

Construct the Confidence Interval

The formula for the confidence interval is: \[ \text{CI} = \bar{x} \pm t^* \times \text{SEM}, \] substituting the values, we get: \[ 24.76 \pm 1.725 \times 1.38 \approx (22.37, 27.15). \] So the 90% confidence interval is approximately (22.37, 27.15) degrees.

06

Interpret the Confidence Interval

We are 90% confident that the true mean HAV angle for the population is between 22.37 degrees and 27.15 degrees.

07

Effect of Including the Outlier

Including the outlier of 50 degrees in the dataset would increase the sample mean and potentially increase the standard deviation, leading to a wider confidence interval. Outliers typically increase variability, affecting the margin of error.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

HAV Angle

Hallux Abducto Valgus (HAV) angle refers to the degree of deformity in the big toe. This angle is crucial in assessing the severity of the deformity, especially among young patients who may require corrective surgery. Measuring the HAV angle involves taking an X-ray and quantifying the level of misalignment in degrees. A larger angle typically indicates more severe deformation, leading to potential discomfort or difficulty in walking.

In the given exercise, the mean HAV angle was recorded at 24.76 degrees for a sample of 21 patients, giving a sense of the average severity among those observed. When reviewing such data, it is important to collect a representative sample to ensure accurate inferences about the broader population of patients with similar conditions.

t-distribution

When we don't know the population standard deviation and the sample size is small (less than 30), we use the t-distribution instead of the normal distribution. This is because the t-distribution accounts for extra variability, or uncertainty, that's inherent with small sample sizes.

In this exercise, the sample size is 21, which fits the criteria for using the t-distribution. The degrees of freedom (df) for a t-distribution is calculated as the sample size minus 1, so here it is 20 (21 - 1). Using the t-distribution table, the critical t-value for a 90% confidence interval with 20 degrees of freedom is approximately 1.725.

This t-value is crucial because it will determine the width of our confidence interval, helping us understand the range in which the true population mean is likely to fall.

Standard Error of the Mean

The Standard Error of the Mean (SEM) gives an estimate of the variation or "spread" of sample means around the population mean. It essentially tells us how much the sample mean would fluctuate if we repeated the same sample selection many times.

To calculate SEM, you divide the sample standard deviation by the square root of the sample size. In our exercise, the standard deviation is 6.34 and the sample size is 21. Thus, SEM is computed as: \[ \text{SEM} = \frac{6.34}{\sqrt{21}} \approx 1.38.\] The SEM is used together with the critical t-value to create the confidence interval for the mean. It helps provide a buffer zone around the sample mean in which we expect to find the true population mean with a specified level of confidence, in this case, 90%.

Outlier Effect

An outlier is an unusual data point that differs significantly from other observations. Outliers can heavily influence statistical measures, like the mean and standard deviation, potentially skewing results.

In this particular exercise, one patient with an HAV angle of 50 degrees was excluded from the analysis due to a measurement issue. If this outlier were included, it would likely elevate the mean HAV angle. Additionally, it would increase the standard deviation, leading to greater variability in the data.

The presence of an outlier could widen the confidence interval, as more variability in the data results in a larger margin of error. This means we would be less certain about the precise location of the true mean within the interval, emphasizing the importance of carefully considering any observed outliers in analyses.