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Understanding how to calculate the sample mean is fundamental in statistics. It represents the average value in a sample data set, providing a quick glimpse into the center of the data. To find the sample mean, you sum up all the values and then divide by the number of observations. The formula is:
\[ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i \]
where \( \bar{x} \) is the sample mean, \( n \) is the number of observations, and \( x_i \) are the individual data points. In the context of the exercise, you added all the ages of the mothers together and then divided by the total count (20) to obtain the sample mean age.

The sample standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the sample mean, while a high standard deviation indicates that the values are spread out over a wider range. To calculate the sample standard deviation (represented as \( s \)), you follow these steps:

• Subtract the sample mean from each data point to find the deviation of each point.
• Square each deviation.
• Add up all the squared deviations.
• Divide by \( n-1 \), where \( n \) is the sample size, to find the variance.
• Take the square root of the variance to get the standard deviation.

The formula is:
\[ s = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2} \]
For the given exercise, you would use this process with the ages of the mothers to find out how much the ages vary from the average (sample mean) age.

The normal distribution, often called the bell curve, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In a normal distribution, the mean, median, and mode are all equal. This distribution is extremely important in statistics because of the central limit theorem, which states that, under many conditions, as sample size increases, the distribution of sample means approaches a normal distribution, regardless of the shape of the original data distribution.
In the case of the exercise, the ages of the mothers are assumed to follow a normal distribution, which allows us to use the Z score and the standard normal distribution table in the subsequent steps to test our hypothesis.

The Z score is a statistical measurement that describes a value's relationship to the mean of a group of values. It is measured in terms of standard deviations from the mean. If a Z score is 0, it indicates that the data point's score is identical to the mean score. Calculating the Z score involves the following formula:
\[ Z = \frac{x - \mu}{\sigma} \]
In this formula, \( x \) is the value from the sample, \( \mu \) is the mean of the population, and \( \sigma \) is the population standard deviation. For sample data, where we use the sample standard deviation (\( s \)), the formula modifies to:
\[ Z = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}} \]
where \( \bar{x} \) is the sample mean, and \( n \) is the sample size. This Z score helps us determine the likelihood of a sample mean occurring within a normal distribution.

Hypothesis testing in statistics is a way to test if there is enough evidence to support a certain belief (alternative hypothesis) about a parameter compared to what has been generally accepted (null hypothesis). The null hypothesis (\( H_0 \)) is a statement of no effect or no difference and is usually the hypothesis that experimenters try to disprove. On the other hand, the alternative hypothesis (\( H_1 \) or (\( H_a \))) expresses a specific claim that there is a significant difference, relationship, or effect.
The process of hypothesis testing involves stating both hypotheses, determining a level of significance (\( \alpha \)), and finding the critical value (Z score or t score) to either reject or fail to reject the null hypothesis. In your exercise, the null hypothesis states that the mean age of mothers of abnormal male children is equal to 28 (\( H_0: \mu = 28 \)), and the alternative hypothesis suggests that the mean age is greater than 28 (\( H_1: \mu > 28 \)).