Investigating Reading Comp. w/ Cultural Capital: A T-Test

[mathmari]

Hey!

The international comparative school performance study PIRLS (Progress in International Reading
Literacy Study) raised in 2011 the reading competences of the fourth graders in more than 40 countries, mong others in Germany.
They would now like to investigate how the cultural capital of the parents affects the reading competencies of the children in the fourth class.
They operationalize the cultural capital as the volume of the books in the parents' house.
They want to find out if there is a significant difference between children whose parents have over 100 books, and children whose parents have a maximum of one hundred books.

A table of statistic is given:
View attachment 6403

where Mittelwert=mean value, Standardabweichung=standard deviation, Standardfehler des Mittelswertes=standard error of mean value. We have to compute a T-Test to find out, if we can confirm out hypothesis and to get a statistic significant relation.
First we have to prove with a F-Test, if we have to apply a double T-Test or a Test of Welch.
At the F-Test and the T-Test we have a significance level of 5 %. How can we check what T-Test we have to apply? (Wondering)

The null hypothesis is that when the parents have more than 100 books then the children are better in reading than others, or not?

Do we get from that that we don't need a double T-Test? (Wondering)
Also using the F-Test how can we check if there is an homogeneity or a heterogeneity of variance? Do we have to use this formula ? (Wondering)

 

[I like Serena]

Hey mathmari! (Smile)

Either way, we're going to do an independent 2-sample t-test.
The only question is how we're going to calculate the standard error SE.
That depends on whether we should assume equal variances (double t-test) or unequal variances (Welch's test).
To figure that out we'll do an F-test, for which we need to calculate the F-score:
$$F=\frac{s_1^2}{s_2^2}$$
We'll have to e.g look up in an F-table what the critical F-score is. (Thinking)EDIT: Oh, and the null hypothesis is (always): there is no difference, everything behaves as if there is nothing special. (Nerd)

 

[mathmari]

We have this table. I haven't really understood at which column and row we have to look? Could you explain it to me? (Wondering)

 

[I like Serena]

We have this http://mathhelpboards.com/advanced-probability-statistics-19/tests-20752.html#post94306. I haven't really understood at which column and row we have to look? Could you explain it to me? (Wondering)

That link links to this page... (Worried)

Anyway, the procedure how to do the F-test for equality of variances is explained here in wiki.

 

[mathmari]

I corrected the link. It holds that the variance is the square of the given standard deviation.

So, $s_1^2=64.93145$ and $s_2^2=70.75413$. Then $F=\frac{s_1^2}{s_2^2}=\frac{64.93145}{70.75413}\approx 0.9177$.

Is this correct? (Wondering)

 

[I like Serena]

I corrected the link. It holds that the variance is the square of the given standard deviation.

So, $s_1^2=64.93145$ and $s_2^2=70.75413$. Then $F=\frac{s_1^2}{s_2^2}=\frac{64.93145}{70.75413}\approx 0.9177$.

Is this correct? (Wondering)

Yes... although it seems we didn't actually square those values... (Worried)

That table is a bit awkward. It doesn't show sufficient degrees of freedom on the x-axis (sample size of the first group minus 1).
Do we have another table? (Wondering)

 

[mathmari]

Yes... although it seems we didn't actually square those values... (Worried)

That table is a bit awkward. It doesn't show sufficient degrees of freedom on the x-axis (sample size of the first group minus 1).
Do we have another table? (Wondering)

$$F=\frac{s_1^2}{s_2^2}=\frac{64.93145^2}{70.75413^2}\approx 0.84218$$ right? (Wondering)

I still haven't understood how to use the table? (Worried)

 

[I like Serena]

$$F=\frac{s_1^2}{s_2^2}=\frac{64.93145^2}{70.75413^2}=\approx 0.84218$$ right? (Wondering)

I still haven't understood how to use the table? (Worried)

Right.

The degrees of freedom are respectively $DF_1 = n_1 -1 = 1541 - 1$ ($x$-axis) and $DF_2 = n_2 - 1= 1598 - 1$ ($y$-axis).
The first table is for the threshold of significance $\alpha=0.05$, which is what we will want.
When we look it up, we find the critical F-value $F^*$. (Nerd)

If $F > F^*$ or $F < \frac{1}{F^*}$, then the variances are significantly different.
Otherwise they are assumed to be equal.
Note that an $F$-value of $1$ means that the variances are exactly the same.

What is the closest $F^*$ value that we can find?
What can we say about on which side of $F^*$ our $F$-value will be? (Wondering)

 

[mathmari]

The degrees of freedom are respectively $DF_1 = n_1 -1 = 1541 - 1$ ($x$-axis) and $DF_2 = n_2 - 1= 1598 - 1$ ($y$-axis).

At a nect question it is asked to compute the degrees of freedom and the choices are:
a) 3137
b) 3129
c) 3345

(Wondering)

The first table is for the threshold of significance $\alpha=0.05$, which is what we will want.
When we look it up, we find the critical F-value $F^*$. (Nerd)

If $F > F^*$ or $F < \frac{1}{F^*}$, then the variances are significantly different.
Otherwise they are assumed to be equal.
Note that an $F$-value of $1$ means that the variances are exactly the same.

What is the closest $F^*$ value that we can find?
What can we say about on which side of $F^*$ our $F$-value will be? (Wondering)

We have that $\frac{1}{0.84218}\approx 1.19$ but at the table the smallest number is $1.93$. What do we do now? (Wondering)

 

[I like Serena]

At a nect question it is asked to compute the degrees of freedom and the choices are:
a) 3137
b) 3129
c) 3345

(Wondering)

That looks as if it's about the degrees of freedom for the t-test we're going to do.
And that depends on whether we're going to do the t-test with equal variances assumed (double t-test) or the t-test with unequal variances assumed (Welch's t-test).
Do we have the formulas for the degrees of freedom in either case? (Wondering)

We have that $\frac{1}{0.84218}\approx 1.19$ but at the table the smallest number is $1.93$. What do we do now? (Wondering)

We find another F-table, since we cannot be quite sure yet whether it is significant or not (although it doesn't look significant)...
Can we find one? (Thinking)

 

[mathmari]

We find another F-table, since we cannot be quite sure yet whether it is significant or not (although it doesn't look significant)...
Can we find one? (Thinking)

At this https://www.ma.utexas.edu/users/davis/375/popecol/tables/f005.html there is 1.19.

 

[I like Serena]

At this https://www.ma.utexas.edu/users/davis/375/popecol/tables/f005.html there is 1.19.

Hmm... that table gives $F^*=1.03$ for $\alpha=0.05$, $DF_1 > 1000$, and $DF_2 > 1000$.
And we had $F=1.19$, so that implies we need to assume unequal variances... (Thinking)

 

[mathmari]

I understand!

To calculate the test statistic do use the formula https://en.wikipedia.org/wiki/Student's_t-test#One-sample_t-test or not?

Is n the sum of the two N's? and which $\overline{X}$ ? (Wondering)

 

[I like Serena]

I understand!

To calculate the test statistic do use the formula https://en.wikipedia.org/wiki/Student's_t-test#One-sample_t-test or not?

Is n the sum of the two N's? and which $\overline{X}$ ? (Wondering)

We need another section on that same page where it says unequal variances... (Thinking)

 

[mathmari]

We need another section on that same page where it says unequal variances... (Thinking)

Oh yes...

We have taht \begin{equation*}t=\frac{\overline{X}_1-\overline{X}_2}{s_{\overline{\Delta}}}\end{equation*} where $\sigma_{\Delta}=\sqrt{\frac{s_1^2}{N_1}+\frac{s_2^2}{N_2}}$.

We have that \begin{equation*}\sigma_{\Delta}=\sqrt{\frac{s_1^2}{N_1}+\frac{s_2^2}{N_2}}=\sqrt{\frac{64.93145^2}{1541}+\frac{70.75413^2}{1598}}= 2.42\end{equation*}

Thereofre we get \begin{equation*}t=\frac{555.3367-512.9353}{2.42}=17.52\end{equation*} right? (Wondering)

 

[I like Serena]

That looks correct yes. (Nod)

 

[mathmari]

That looks correct yes. (Nod)

Great! Thank you very much! (Happy)