An organization is studying whether women have caught up to men in starting pay after attending college. They randomly sampled 28 women, who earned an average of $38,293.78 out of college with a standard deviation of $5,170.22. Twenty-four men were also randomly sampled, and their earnings averaged $41,981.82 with a standard deviation of $3,195.42. Using a significance level of 0.02, do these data provide strong evidence that women have not yet caught up to men in terms of pay? If so, can we make a causal con- clusion? If so, explain why. If not, provide an example of why the causal interpretation would not be valid.
Z = (mean1 - mean2)/standard error (SE) of difference between means
SEdiff = √(SEmean1^2 + SEmean2^2)
SEm = SD/√n
If only one SD is provided, you can use just that to determine SEdiff.
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to the Z score.
The probability of being a smoker for a population of college students is 0.20.The standard deviation for samples of 1600 students is 0.01. The standard deviation would be smallest for which of these sample sizes?
A) 16
To determine if these data provide strong evidence that women have not yet caught up to men in terms of pay, we can perform a hypothesis test. Let's go through the steps:
Step 1: State the null and alternative hypotheses:
- Null hypothesis (H0): Women and men have the same average starting pay after attending college.
- Alternative hypothesis (Ha): Women have a lower average starting pay than men after attending college.
Step 2: Determine the significance level (α):
The significance level provided in the question is 0.02.
Step 3: Select the appropriate test statistic:
Since we are comparing the means of two independent samples, and the sample sizes are relatively small (<30), we can use a t-test. This is known as the independent samples t-test.
Step 4: Calculate the test statistic:
Using the following formula for an independent samples t-test:
t = (mean1 - mean2) / sqrt((s1^2/n1) + (s2^2/n2))
where:
mean1 = average earnings of women
mean2 = average earnings of men
s1 = standard deviation of women's earnings
s2 = standard deviation of men's earnings
n1 = number of women sampled
n2 = number of men sampled
Plugging in the values:
mean1 = $38,293.78
mean2 = $41,981.82
s1 = $5,170.22
s2 = $3,195.42
n1 = 28
n2 = 24
t = (38,293.78 - 41,981.82) / sqrt((5,170.22^2/28) + (3,195.42^2/24))
Step 5: Determine the degrees of freedom:
The degrees of freedom (df) for an independent samples t-test can be computed using the following formula:
df = (s1^2/n1 + s2^2/n2)^2 / (((s1^2/n1)^2 / (n1 - 1)) + ((s2^2/n2)^2 / (n2 - 1)))
Plugging in the values:
df = ( (5,170.22^2/28) + (3,195.42^2/24) )^2 / ( ( (5,170.22^2/28)^2 / (28-1) ) + ( (3,195.42^2/24)^2 / (24-1) ) )
Step 6: Determine the critical value:
Based on the given significance level (α = 0.02) and the degrees of freedom, we can obtain the critical value from the t-distribution table.
Step 7: Compare the test statistic and the critical value:
If the absolute value of the test statistic is greater than the critical value, we would reject the null hypothesis. This would provide evidence that women have not yet caught up to men in terms of pay.
Step 8: Make a conclusion:
If the null hypothesis is rejected, we can conclude that there is strong evidence that women have not caught up to men in terms of pay.
However, regarding the possibility of making a causal conclusion, it is important to note that this study is observational, not experimental. The organization simply randomly sampled individuals and collected data. Therefore, we cannot establish a causal relationship between attending college and starting pay. Other factors, such as job selection, negotiations, and personal choices, may also influence the pay disparity. Thus, we can only conclude there is an association or correlation between attending college and pay, but not necessarily a causal relationship.