A teacher gives a reading skills test to a third-grade class of n = 25 at the beginning of the
school year. To evaluate the changes that occur during the year, students are tested again
at the end of the year. Their test scores showed an average improvement of MD = 12.7
points with s2
= 100.
State the null hypothesis.
b. Compute the necessary hypothesis test using a two-tailed test with = .05. Show
your calculations.
c. Make a decision about your null hypothesis. (Include the critical value you used to
base your decision.)
d. Write an APA format statement summarizing your findings (include a measure of
effect size if necessary.)
a. The null hypothesis for this scenario would be that there is no significant difference in the reading skills test scores between the beginning and the end of the year for the third-grade class.
b. To compute the necessary hypothesis test, we will use a two-tailed test with α = 0.05.
The formula for a two-tailed hypothesis test is as follows:
t = (sample mean - population mean) / (sample standard deviation / √sample size)
Given that the sample mean improvement is MD = 12.7 points, the population mean improvement is assumed to be zero (since it's the null hypothesis), and the sample standard deviation is s = √s2 = 10 (since s2 = 100), and the sample size is n = 25.
Plugging the values into the formula, we have:
t = (12.7 - 0) / (10 / √25)
t = 12.7 / (10 / 5)
t = 12.7 * 5 / 10
t = 63.5 / 10
t = 6.35
c. To make a decision about the null hypothesis, we compare the calculated t-value (6.35) with the critical t-value. Since the test is two-tailed and α = 0.05, we divide α by 2 to get α/2 = 0.05/2 = 0.025.
Using a t-table or a statistical software, we find the critical t-value for a two-tailed test with α = 0.025 and 24 degrees of freedom (n - 1 = 25 - 1 = 24) to be approximately ±2.064.
Since 6.35 > 2.064, the calculated t-value falls in the rejection region. This means that the observed improvement in the test scores is statistically significant.
d. An APA format statement summarizing the findings could be:
"The third-grade students in our study showed a significant improvement in their reading skills test scores from the beginning to the end of the school year (t(24) = 6.35, p < .05). The average improvement was 12.7 points, indicating a substantial effect size."
a. The null hypothesis for this study is that there is no significant improvement in the reading skills of the third-grade class at the end of the year compared to the beginning.
b. To test the null hypothesis, we can use the formula for a t-test:
t = (MD - μd) / (s / √n)
where MD is the mean difference, μd is the population mean difference (assumed to be 0 under the null hypothesis), s is the standard deviation of the sample differences, and n is the sample size.
In this case, MD = 12.7, s2 = 100, and n = 25.
First, calculate the standard deviation using the formula:
s = √s2 = √100 = 10
Now, calculate the t-value using the formula:
t = (12.7 - 0) / (10 / √25) = 12.7 / (10 / 5) = 12.7 / 2 = 6.35
c. To make a decision about the null hypothesis, we compare the calculated t-value to the critical t-value at the desired significance level α = 0.05. In a two-tailed test, α is split between the rejection regions in both tails.
Looking up the critical t-value in a t-distribution table with 24 degrees of freedom at α = 0.05 two-tailed, we find the critical t-value to be approximately ±2.064.
Since the calculated t-value of 6.35 is outside the critical region of ±2.064, we reject the null hypothesis.
d. The APA format statement summarizing the findings could be:
"The reading skills of a third-grade class showed a significant improvement at the end of the school year compared to the beginning, t(24) = 6.35, p < .05. The average improvement in test scores was 12.7 points (M = 12.7, SD = 10)."
Z = MD/SEm
SEm = SD/√n
Online "^" is used to indicate an exponent, e.g., x^2 = x squared.
SD = √s^2
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z score.
You can take it from there.