Question

A student set out to see if female students spend more time studying than male students. He randomly surveyed 30 females and 30 males and asked them how much time they spend studying, on average, per day. The data is obtained is shown below.

H0: mean study time for females = mean study time for males

H1: mean study time for females > mean study time for males

1. What type of test should we use to analyze this data and why?

2. Are there any assumptions that we must make about the two populations (female students / male students) before we proceed?

3. Analyze the data with the appropriate test using a 10% significance level. State the p-value and compare it to alpha.

4. Which hypothesis should be supported?

5. Could you have made an error in your decision? If so, what? And what could have caused you to make an error?

# of hours spent studying	female counts	male counts
1	6	10
2	7	6
3	7	7
4	2	5
5	3	1
6	3	0
7	2	1

Answer 1

1.What type of test should we use to analyze this data and why?

We use t-test of difference of means.

This is because:

1.since the sample size is small

2. samples are independent.

2.Are there any assumptions that we must make about the two populations (female students / male students) before we proceed?

The assumptions of the two-sample t-test are:

1. The data are continuous (not discrete).

2. The data follow the normal probability distribution.

3. The variances of the two populations are equal.

4. The two samples are independent. There is no relationship between the individuals in one sample as compared to the other (as there is in the paired t-test).

5. Both samples are simple random samples from their respective populations. Each individual in the population has an equal probability of being selected in the sample

3.Analyze the data with the appropriate test using a 10% significance level. State the p-value and compare it to alpha.

first we calculate mean ans dtandard deviation for both the samples:

	Female	Female2
	6	36
	7	49
	7	49
	2	4
	3	9
	3	9
	2	4
Sum =	30	160

The sample mean is computed as follows:

Also, the sample variance is

Therefore, the sample sandartd deviation is

	Male	Male2
	10	100
	6	36
	7	49
	5	25
	1	1
	0	0
	1	1
Sum =	30	212

The sample mean is computed as follows:

Also, the sample variance is

Therefore, the sample sandartd deviation is

The provided sample means are shown below:

Xˉ1​=4.286

Xˉ2​=4.286

Also, the provided sample standard deviations are:

s1​=2.289

s2​=3.729

and the sample sizes are n1​=7 and n2​=7.

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: μ1​ = μ2​ or mean study time for females = mean study time for males

Ha: μ1​ > μ2​ or mean study time for females > mean study time for males

This corresponds to a right-tailed test, for which a t-test for two population means, with two independent samples, with unknown population standard deviations will be used.

(2) Rejection Region

Based on the information provided, the significance level is α=0.10, and the degrees of freedom are df=12. In fact, the degrees of freedom are computed as follows, assuming that the population variances are equal:

Hence, it is found that the critical value for this right-tailed test is tc​=1.356, for α=0.10 and df=12.

The rejection region for this right-tailed test is R={t:t>1.356}.

(3) Test Statistics

Since it is assumed that the population variances are equal, the t-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that t=0≤tc​=1.356, it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value is p=0.5, and since p=0.5≥0.10, it is concluded that the null hypothesis is not rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population mean μ1​ is greater than μ2​, at the 0.10 significance level Hence,mean study time for females = mean study time for males

4.Which hypothesis should be supported?

NULL HYPOTHESIS.

5.Could you have made an error in your decision? If so, what? And what could have caused you to make an error?

while a type II error is the non-rejection of a false null hypothesis (also known as a "false negative" finding or conclusion).

Hence we made an type II error.Here a researcher concludes there is not a significant effect, when actually there really is.

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