Final answer:
In statistics, a z-score represents the number of standard deviations a data point is from the mean, which is calculated using (X - μ) / σ. For small sample sizes or when the population standard deviation is unknown, the Student's t-distribution is used for hypothesis testing. The normal distribution is used when the population standard deviation is known and the sample size is adequate.
Step-by-step explanation:
When a student is seeking to understand the probability of obtaining a certain test score or test result, they are delving into the subject of statistics, a branch of mathematics. Specifically, we are discussing the calculation of a z-score and the selection of the appropriate probability distribution for a hypothesis test based on sample data in relation to an assumed population parameter. In the case that the population standard deviation is known and the sample size is large enough (typically n > 30 is considered sufficient), the normal distribution can be used. However, with smaller sample sizes and if the population standard deviation is unknown, the Student's t-distribution is most appropriate.
For example, to calculate a z-score given a data set with a mean (μ) of 5 and standard deviation (σ) of 2, where we want to find the number of standard deviations a score of 11 is above the mean, the formula is z = (X - μ) / σ. In this case, (11 - 5) / 2 = 3, meaning that a score of 11 is 3 standard deviations above the mean.
In hypothesis testing scenarios like the Stanford-Binet IQ test, where a claim is made that the mean is greater than a certain value and the sample is relatively small (for example, less than 30), the Student's t-distribution is the correct distribution to use. The t-distribution accounts for the additional uncertainty due to estimating the population standard deviation from the sample, especially when the sample size is small.