The value of the z-score that is obtained for a hypothesis test is influenced by several factors. Some factors influence the size of the numerator of the z-score and other factors influence the size of the standard error in the denominator.
For each of the following indicate whether the factor influences the numerator or denominator of the z-score and determine whether the effect would be to increase the value of z(farther from zero) or decrease the value of z (closer to zero).
In each case, assume that all other components of the z-score remain constant.
a)increase the sample size
b) increase the population standard deviation
c)increase the difference between the sample mean and the value of µ specified in the null hypothesis
Z = (mean - ƒÊ)/SE
Anything that increases the numerator, increases the value of Z. Anything that increases the value of the denominator, decreases the value of Z.
a) since a larger sample would decrease the SE, that would increase Z.
b) since the increase in population SD would increase the SE, it would decrease Z.
c) since the difference is in the numerator, it would increase Z.
I hope this helps. Thanks for asking.
a) Increasing the sample size would influence the denominator of the z-score. Specifically, it would decrease the standard error, which is the denominator of the z-score formula. This would result in a smaller standard error and therefore a larger z-score, moving the value of z farther from zero.
b) Increasing the population standard deviation would also influence the denominator of the z-score. Specifically, it would increase the standard error and therefore decrease the z-score, moving the value of z closer to zero.
c) Increasing the difference between the sample mean and the value of µ specified in the null hypothesis would influence the numerator of the z-score. Specifically, it would increase the numerator, increasing the value of z and moving it farther from zero.
To determine whether a factor influences the numerator or denominator of the z-score, we need to understand the components of the z-score formula. The z-score is calculated as:
z = (x - μ) / (σ / √n)
Where:
- x is the sample mean
- μ is the population mean specified in the null hypothesis
- σ is the population standard deviation
- n is the sample size
Now, let's analyze each factor:
a) Increase the sample size:
Increasing the sample size (n) will affect the denominator of the z-score equation. As you can see from the formula, the term σ / √n is in the denominator. When the sample size increases, the denominator gets larger, leading to a smaller standard error. Consequently, it decreases the value of the z-score (closer to zero).
b) Increase the population standard deviation:
Increasing the population standard deviation (σ) affects the denominator of the z-score equation as well. As the population standard deviation rises, the denominator becomes larger, resulting in a smaller standard error. Hence, the value of the z-score decreases (closer to zero).
c) Increase the difference between the sample mean and the value of µ specified in the null hypothesis:
Increasing the difference between the sample mean (x) and the value of μ specified in the null hypothesis affects the numerator of the z-score equation. Specifically, the term (x - μ) in the numerator gets larger. As a result, the value of the z-score increases (farther from zero).
In summary:
- Increasing the sample size (n) and the population standard deviation (σ) both decrease the value of the z-score (closer to zero).
- Increasing the difference between the sample mean (x) and the value of μ specified in the null hypothesis increases the value of the z-score (farther from zero).