Question

Please provide the following information for Problems. (a) What is the level of significance? State the null and alternate hypotheses. (b) Check Requirements What sampling distribution will you use? What assumptions are you making? Compute the sample test statistic and corresponding $z$ or $t$ value as appropriate. (c) Find (or estimate) the $P$ -value. Sketch the sampling distribution and show the area corresponding to the $P$ -value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level $\alpha ?$ (e) Interpret your conclusion in the context of the application. Note: For degrees of freedom $d . f .$ not in the Student's $t$ table, use the closest $d . f .$ that is smaller. In some situations, this choice of $d . f .$ may increase the $P$ -value a small amount and therefore produce a slightly more "conservative" answer. Medical: REM Sleep REM (rapid eye movement) sleep is sleep during which most dreams occur. Each night a person has both REM and non-REM sleep. However, it is thought that children have more REM sleep than adults (Reference: Secrets of Sleep by Dr. A. Borbely). Assume that REM sleep time is normally distributed for both children and adults. A random sample of $n_{1}=10$ children (9 years old) showed that they had an average REM sleep time of $\bar{x}_{1}=2.8$ hours per night. From previous studies, it is known that $\sigma_{1}=0.5$ hour. Another random sample of $n_{2}=10$ adults showed that they had an average REM sleep time of $\bar{x}_{2}=2.1$ hours per night. Previous studies show that $\sigma_{2}=0.7$ hour. Do these data indicate that, on average, children tend to have more REM sleep than adults? Use a $1 \%$ level of significance.

Name: Problem 15, Chapter 8: Understandable Statistics, Concepts and Methods
Uploaded: 2021-11-18T20:51:26-08:00
Duration: 3 min 6 s
Channel: Sheryl Ezze

   Please provide the following information for Problems.
(a) What is the level of significance? State the null and alternate hypotheses.
(b) Check Requirements What sampling distribution will you use? What assumptions are you making? Compute the sample test statistic and corresponding $z$ or $t$ value as appropriate.
(c) Find (or estimate) the $P$ -value. Sketch the sampling distribution and show the area corresponding to the $P$ -value.
(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level $\alpha ?$
(e) Interpret your conclusion in the context of the application. Note: For degrees of freedom $d . f .$ not in the Student's $t$ table, use the closest $d . f .$ that is smaller. In some situations, this choice of $d . f .$ may increase the $P$ -value a small amount and therefore produce a slightly more "conservative" answer.
Medical: REM Sleep REM (rapid eye movement) sleep is sleep during which most dreams occur. Each night a person has both REM and non-REM sleep. However, it is thought that children have more REM sleep than adults (Reference: Secrets of Sleep by Dr. A. Borbely). Assume that REM sleep time is normally distributed for both children and adults. A random sample of $n_{1}=10$ children (9 years old) showed that they had an average REM sleep time of $\bar{x}_{1}=2.8$ hours per night. From previous studies, it is known that $\sigma_{1}=0.5$ hour. Another random sample of $n_{2}=10$ adults showed that they had an average REM sleep time of $\bar{x}_{2}=2.1$ hours per night. Previous studies show that $\sigma_{2}=0.7$ hour. Do these data indicate that, on average, children tend to have more REM sleep than adults? Use a $1 \%$ level of significance.

Understandable Statistics, Concepts and Methods

Charles Henry Brase,… 12th Edition

Chapter 8, Problem 15

Instant Answer

Solved by Expert Sheryl Ezze

11/18/2021

Step 1

The null hypothesis is that the average REM sleep time for children is equal to that of adults, while the alternate hypothesis is that the average REM sleep time for children is greater than that of adults. In mathematical terms, this can be written as: Null Show more…

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Please provide the following information for Problems. (a) What is the level of significance? State the null and alternate hypotheses. (b) Check Requirements What sampling distribution will you use? What assumptions are you making? Compute the sample test statistic and corresponding $z$ or $t$ value as appropriate. (c) Find (or estimate) the $P$ -value. Sketch the sampling distribution and show the area corresponding to the $P$ -value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level $\alpha ?$ (e) Interpret your conclusion in the context of the application. Note: For degrees of freedom $d . f .$ not in the Student's $t$ table, use the closest $d . f .$ that is smaller. In some situations, this choice of $d . f .$ may increase the $P$ -value a small amount and therefore produce a slightly more "conservative" answer. Medical: REM Sleep REM (rapid eye movement) sleep is sleep during which most dreams occur. Each night a person has both REM and non-REM sleep. However, it is thought that children have more REM sleep than adults (Reference: Secrets of Sleep by Dr. A. Borbely). Assume that REM sleep time is normally distributed for both children and adults. A random sample of $n_{1}=10$ children (9 years old) showed that they had an average REM sleep time of $\bar{x}_{1}=2.8$ hours per night. From previous studies, it is known that $\sigma_{1}=0.5$ hour. Another random sample of $n_{2}=10$ adults showed that they had an average REM sleep time of $\bar{x}_{2}=2.1$ hours per night. Previous studies show that $\sigma_{2}=0.7$ hour. Do these data indicate that, on average, children tend to have more REM sleep than adults? Use a $1 \%$ level of significance.

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Key Concepts

Decision Rule

The decision rule in hypothesis testing involves comparing the P-value to the predetermined level of significance. If the P-value is less than or equal to the level of significance, the null hypothesis is rejected, indicating statistically significant results. Conversely, if the P-value exceeds the significance level, there is insufficient evidence to reject the null hypothesis.

P-value Interpretation

The P-value measures the probability of obtaining a test statistic at least as extreme as the one observed, assuming that the null hypothesis is true. It provides a gauge for the strength of the evidence against the null hypothesis. A small P-value indicates that such an extreme result is unlikely under the null hypothesis, thereby supporting the alternative hypothesis.

Test Statistic Calculation

The test statistic is a standardized value calculated from sample data, which is used to assess the evidence against the null hypothesis. Its computation involves comparing the observed sample statistic to the expected value under the null hypothesis, divided by the standard error. Depending on whether the population standard deviation is known or estimated from the sample, a z-test or a t-test statistic will be used.

Contextual Interpretation

Contextual interpretation refers to explaining the results of a statistical test in terms that relate back to the research question or practical application. It involves describing what rejecting or not rejecting the null hypothesis means within the real-world setting, ensuring that statistical conclusions are made meaningful in the context of the study’s subject matter.

Sampling Distribution

A sampling distribution is the probability distribution of a given statistic based on a random sample. In hypothesis testing, it provides the framework for determining the likelihood of observing data similar to the sample under the null hypothesis. When population parameters such as the standard deviation are known, and the sample size is sufficiently large or the underlying population is normal, the sampling distribution will typically be assumed to follow a normal distribution.

Level of Significance

The level of significance is a threshold probability used to decide when to reject the null hypothesis. It quantifies the risk of making a Type I error, which occurs when a true null hypothesis is incorrectly rejected. This level, often denoted by ?, is set prior to analysis, ensuring that conclusions about statistical significance are made consistently.

Null and Alternative Hypotheses

This concept involves formulating a formal statement to test for an effect or difference. The null hypothesis (H0) generally represents a position of no change or no difference and is assumed true until evidence indicates otherwise. The alternative hypothesis (H1) reflects the claim or effect that the study seeks to support, providing a basis for concluding whether there is a significant difference between groups or conditions.

Assumptions

Assumptions in statistical testing refer to the conditions believed to be true for the analysis to be valid. Common assumptions include independence of observations, normality of the data distribution, and, when applicable, the known or equal population variances. Meeting these assumptions is crucial for the accuracy of the inference drawn from the statistical test.

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