10.2 Two column model step by step example of a hypothesis test

• State the question: State what we want to determine and what level of significance is important in your decision.

We are asked to test the hypothesis that the mean time boys and girls between 7 and 11 spend playing sports each day is the same. We do not know the population standard deviations. The significance level is 5%.

• Plan: Based on the above question(s) and the answer to the following questions, decide which test you will be performing. Is the problem about numerical or categorical data?If the data is numerical is the population standard deviation known? Do you have one group or two groups?What type of model is this?

We have bivariate, quantitative data. We have two independent groups. We have a sample of 9 for the girls and 16 for the boys. We do not know the population standard deviations. Therefore, we can perform a Students t-Test for independent samples, with approximately 18.8462 degrees of freedom. Our model will be:

• Hypothesis: State the null and alternative hypotheses in words and then in symbolic form
• 1. Express the hypothesis to be tested in symbolic form.
• 2. Write a symbolic expression that must be true when the original claim is false.
• 3. The null hypothesis is the statement which included the equality.
• 4. The alternative hypothesis is the statement without the equality.

Null hypothesis in words: The null hypothesis is that the true mean time playing sports each day of girls is equal to the true mean time each day of boys playing sports.
Null Hypothesis symbolically: H ₀ : Mean time μ _G = μ _B
Alternative Hypothesis in words: The alternative is that the true mean time playing sports each day of girls is Not equal to the true mean time each day of boys playing sports.
Alternative Hypothesis symbolically: H _a : Mean time μ _G ≠ μ _B

• The criteria for the inferential test stated above: Think about the assumptions and check the conditions.If your assumptions include the need for particular types of data distribution, please insert the appropriate graphs or charts if necessary.

Randomization Condition: The samples are random samples.
Independence Assumption: It is reasonable to think that the times within the samples are independent when you have a random samples. There is no reason to think the time spent on sports of one child has any bearing on the time spent on sports of another child.
Independent Groups Assumption: It is reasonable to think that the boys and girls times are independent of each other.
10% Condition: I assume the number of children in the community where this was done is more than 250, so 9 girl times and 16 boy times is less than 10% of each population.
Nearly Normal Condition: The problem states that both are from normal populations.
Sample Size Condition: Since the distribution of the times are both normal, my samples of 9 and 16 scores are large enough.

Compute the test statistic:

$\overline{X_G}=\; 2;S_G=\sqrt{0.75};\; n\; =\; 9\; ;\overline{X_B}=3.2;S_G=1.00;$

• $n\; =\; 16;SE\; =\sqrt{\frac{S^{2_{}}}{n_{\mathrm{G}}}+\frac{S^{2_{}}}{n_{\mathrm{B}}}}+\sqrt{\frac{.75}{9}+\frac{1}{16}}=\sqrt{0.1458}=\; 0.382;df\; \approx \; 18.8462$

• $(\overline{X_G}-\overline{X_B})\; =\; 2\; -\; 3.2\; =\; -1.2$

$t\; =\frac{(\overline{X_G}-\overline{X_B})\; -\; (0)}{SE}=\frac{2\; -\; 3.2}{0.382}=\; -3.142$

Sketch the test statistic and critical region:

Determine the P-value

State whether you reject or fail to reject the Null hypothesis.

If you reject the null hypothesis, continue to complete the following

Calculate and display your confidence interval for the Alternative hypothesis.

State your conclusion based on your confidence interval.