to evaluate the sample mean against the hypothesized population mean, we use the concept of z scores to determine how different the two means are.
To evaluate the sample mean against the hypothesized population mean, we use the concept of z-scores to determine how different the two means are.
When comparing a sample mean to a hypothesized population mean, z-scores are used to assess the standard deviation of the sample mean from the population mean. This process helps in understanding the significance of the difference between the sample mean and the population mean.
The formula to calculate the z-score is:
z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}
Where:
- \bar{x} = sample mean
- \mu = population mean
- \sigma = standard deviation of the population
- n= sample size
The z-score indicates how many standard deviations the sample mean is from the population mean. A positive z-score means the sample mean is above the population mean, while a negative z-score indicates the sample mean is below the population mean.
The z-score can then be used to find the probability of obtaining a sample mean at least as extreme as the one observed, assuming the null hypothesis (the hypothesized population mean) is true. This probability is found using a z-table or a statistical software package.
In summary, z-scores are a fundamental tool in statistics for comparing sample means to hypothesized population means and determining the significance of the difference between the two means.