Question

As part of the same study described in Exercise 6.254 , the researchers also were interested in whether babies preferred singing or speech. Forty-eight of the original fifty infants were exposed to both singing and speech by the same woman. Interest was again measured by the amount of time the baby looked at the woman while she made noise. In this case the mean time while speaking was 66.97 with a standard deviation of \(43.42,\) and the mean for singing was 56.58 with a standard deviation of 31.57 seconds. The mean of the differences was 10.39 more seconds for the speaking treatment with a standard deviation of 55.37 seconds. Perform the appropriate test to determine if this is sufficient evidence to conclude that babies have a preference (either way) between speaking and singing.

Short answer

To make a definitive statement, the test statistic should be calculated and compared with the critical value from the t-distribution table. Depending on whether it is less or greater than the critical value, a decision can be made whether to reject or not reject the null hypothesis. Nevertheless, given the relatively high standard deviation in the speaking treatment condition, it may be suggestive that the null hypothesis cannot be rejected, signifying that there might not be a significant difference in the babies' preference for speech or song. More steps involved in the computation process are required to reach a more accurate conclusion.

Step-by-step solution

State the null and alternative hypotheses

The null hypothesis \(H_0\) asserts that there is no difference between the average durations babies listen to speaking and singing. Stated in mathematical terms, the null hypothesis is \(H_0: \mu_{speak} = \mu_{sing}\). The alternative hypothesis \(H_a\) suggests that there is a significant difference between these two variables. It is \(H_a: \mu_{speak} \neq \mu_{sing}\).

Choose the level of significance

The level of significance has not been given. However, 0.05 is a common standard in most scientific studies and will be used here. So, \(\alpha = 0.05\)

Compute the test statistic

To perform the t-test for independent samples, the test statistic is calculated by the formula \n\[ t = \frac{\bar{x}_1 - \bar{x}_2}{s_{d} / \sqrt{n}}\] \nwhere \n\(\bar{x}_1, \bar{x}_2 \) are the means of two samples, \n\(s_{d}\) is the standard deviation of differences between two samples, \nand n is the number of observations. \nInputting the given values into the formula, \n\n \[ t = \frac{10.39}{55.37 / \sqrt{48}} \]

Determine the critical value

The critical value for a two-tailed t-test with \(\alpha = 0.05\) and degrees of freedom \(df = n - 1 = 47\) is approximately ±2.0116 according to t-distribution tables.

Make a decision

If the absolute value of the test statistic from step 3 is greater than the critical value from step 4, we reject \(H_0\). Otherwise, we do not reject \(H_0\).

Key concepts

Null and Alternative Hypotheses

Understanding the difference between null and alternative hypotheses is crucial in hypothesis testing. The null hypothesis, denoted as \(H_0\), represents the default or status quo assumption that there is no effect or no difference in the general population. For the exercise in question, the null hypothesis posits that there is no preference between speech and singing for babies, which mathematically can be expressed as \(H_0: \mu_{speak} = \mu_{sing}\). On the flip side, the alternative hypothesis, denoted as \(H_a\) or \(H_1\), is the assertion that contradicts the null hypothesis. It's what the researcher aims to support. In this case, the alternative hypothesis claims there is a preference, either for speaking or singing, hence \(H_a: \mu_{speak} eq \mu_{sing}\). Choosing the right hypotheses sets the stage for the entire hypothesis testing process.

Significance Level

The significance level, often denoted by the Greek letter \(\alpha\), gauges the threshold at which the null hypothesis is rejected in favor of the alternative. A commonly used significance level is 0.05, which indicates a 5% risk of concluding that a difference exists when there is none (type I error). Selecting the appropriate significance level affects how stringent the hypothesis test is and impacts the conclusions drawn from the data. In the exercise, a standard significance level of \(\alpha = 0.05\) is selected. This level of significance is a balance between being too lenient and too strict, allowing for a fair assessment of the evidence against the null hypothesis.

T-Test for Independent Samples

The t-test for independent samples compares the means of two independent groups to determine whether there is statistical evidence that the associated population means are significantly different. It's ideal for scenarios like the exercise where two different conditions (speaking and singing) are tested on different occasions. The formula for the t-statistic in an independent samples t-test is \[ t = \frac{\bar{x}_1 - \bar{x}_2}{s_{d} / \sqrt{n}} \] where \(\bar{x}_1\) and \(\bar{x}_2\) are the sample means for each group, \(s_{d}\) is the standard deviation of the differences, and \(n\) is the sample size. A critical aspect is to verify that the samples are independent and the data are approximately distributed normally.

Test Statistic

The test statistic is a standardized value used to decide whether to reject the null hypothesis. It's the result of transforming the observed data into a single value that can be compared against the critical value from a statistical distribution. In the context of the t-test for independent samples, the test statistic quantifies the distance between the sample means relative to the variability in the data. The calculated t-statistic provides a means of determining how extreme the observed result is, assuming that the null hypothesis is true. In our exercise, it's derived from the formula mentioned above.

Degrees of Freedom

Degrees of freedom, often abbreviated as df, are a concept tied to the precision with which a statistic estimates a parameter. It reflects the number of independent values that can vary in the calculation of a statistic. In a t-test, the degrees of freedom are typically the number of observations minus the number of parameters that need to be estimated. For an independent samples t-test, the degrees of freedom are calculated as \(df = n_1 + n_2 - 2\) where \(n_1\) and \(n_2\) represent the sample sizes of the two groups. In the exercise example, the degrees of freedom are 47, which comes from 48 observations minus 1.

T-distribution

The t-distribution is a probability distribution that arises when estimating the mean of a normally distributed population in situations where the sample size is small and the population standard deviation is unknown. It's bell-shaped and symmetric like the normal distribution but has heavier tails, meaning that it gives more probability to observations further from the mean. This shape allows for the increased variability expected with smaller sample sizes. In hypothesis testing, the t-distribution helps to find the critical value that the test statistic should exceed to reject the null hypothesis. For the exercise with 47 degrees of freedom, one would refer to the t-distribution to find the appropriate critical value, given the chosen significance level.