Question

Gene mutations have been found in patients with muscular dystrophy. In one study, it was found that there were defects in the gene coding of sarcoglycan proteins in 23 of 180 patients with limb-girdle muscular dystrophy. \({ }^{19}\) Use these data to construct a \(99 \%\) confidence interval for the corresponding population proportion.

Short answer

The 99% confidence interval for the population proportion is (0.0921, 0.1787).

Step-by-step solution

Identify the sample proportion

The sample proportion (p-hat) is the ratio of the number of patients with sarcoglycan gene defects to the total number of patients studied. It is calculated as p-hat = x/n, where x is the number of successes and n is the total sample size.

Calculate the sample proportion

For this study, x = 23 (number of patients with gene defects) and n = 180 (total number of patients). Therefore, p-hat = 23/180.

Determine the z-value for a 99% confidence level

A 99% confidence level corresponds to a 0.01 significance level (alpha). Since the normal distribution is symmetric, we look for a z-value such that 0.5% of the distribution lies to the right, which for a standard normal distribution corresponds to a z-value of approximately 2.576.

Calculate the standard error of the proportion

The standard error (SE) of the proportion is given by the formula SE = sqrt((p-hat * (1 - p-hat)) / n).

Calculate the margin of error

The margin of error (E) for the confidence interval is calculated as E = z * SE, where z is the z-value from Step 3.

Construct the confidence interval

The 99% confidence interval for the population proportion is then p-hat ± E.

Compute the confidence interval

Substitute the values of p-hat, z, and SE into the confidence interval formula to obtain the lower and upper limits of the interval.

Key concepts

Gene Mutations in Muscular Dystrophy

Muscular dystrophy is a group of genetic diseases characterized by progressive muscle weakness and degeneration. The muscles affected include skeletal muscles, which are responsible for movement, as well as possibly heart and respiratory muscles. One of the paths to understanding these diseases involves studying gene mutations, such as defects in the sarcoglycan proteins as found in some patients.

These sarcoglycan proteins are part of a larger protein complex that helps stabilize muscle tissue during contraction. When mutations occur, this complex can be disrupted, leading to the symptoms associated with limb-girdle muscular dystrophy. Conducting research on populations with these mutations provides invaluable data that aids in the development of treatments and may also help predict the inheritance patterns of the disease.

Sample Proportion Calculation

When conducting studies, researchers often deal with a sample of the population rather than the entire group. The sample proportion, represented by \( \hat{p} \), is a fundamental statistic in such studies, referring to the ratio of subjects in the sample with a particular attribute to the total sample size. For example, in the study of patients with limb-girdle muscular dystrophy, the ratio of patients with sarcoglycan gene defects to the total number of patients observed is termed the sample proportion. It is a crucial step in analyzing data, as it sets the stage for further statistical inference such as confidence intervals.

Standard Error of the Proportion

The standard error of the proportion is a measurement that captures the variability of the sample proportion. It quantifies how much we would expect the proportion to vary if we were to draw numerous samples from the same population.

The formula for standard error (SE) is \( SE = \sqrt{\frac{\hat{p} (1 - \hat{p})}{n}} \), where \( \hat{p} \) is the sample proportion and \( n \) is the total number of observations in the sample. A smaller standard error typically suggests that the sample proportion is a more reliable estimate of the population proportion, making it a critical factor when reporting the precision of study findings.

Z-value for Confidence Level

Confidence intervals are a range of values used to estimate the true population proportion, and they are constructed using a confidence level. This level represents the degree of certainty we have that the interval contains the population parameter. To compute the interval, we use a z-value, which is a point on the standard normal distribution.

The z-value for a 99% confidence level signifies that we are 99% confident the true population proportion lies within our calculated interval. The z-value corresponds to the point on a standard normal curve where the tail, the portion beyond that value, contains only 0.5% of the distribution due to the curve's symmetry. For a 99% level, this z-value is approximately 2.576, indicating that our margin of error will help ensure that 99% of similar sample proportions would also fall into this interval if we repeated the study many times under the same conditions.