When people learn a new task, their performance usually improves when they are tested the next day, but only if they get 6 hours sleep (Sticckgold, et al., 2000). The following data demonstrate this phenomenon. The participants learned a visual discrimination task on one day. Half of the participants were allowed to have at least 6 hours of sleep and the other half were kept awake all night. 6 hours sleep No sleep n =14 n = 14 M = 72 M =65 SS = 932 SS = 706 Is there a significant difference between the two conditions? Use a two-tailed test with α = .01. Remember to use the attached t-distribution chart when obtaining your critical region. Please answer the question using all of the steps: null in word, alternative in words, null in symbols, alternative in symbols, critical region t, df, all steps in the analysis computing your independent computed t, make a decision, and give a conclusion.

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Description: When people learn a new task, their performance usually improves when they are tested the next day, but only if they get 6 hours sleep (Sticckgold, et al., 2000). The following data demonstrate this phenomenon. The participants learned a visual disc

Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018

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Section10.6: Summarizing Categorical Data

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When people learn a new task, their performance usually improves when they are tested the next day, but only if they get 6 hours sleep (Sticckgold, et al., 2000). The following data demonstrate this phenomenon. The participants learned a visual discrimination task on one day. Half of the participants were allowed to have at least 6 hours of sleep and the other half were kept awake all night.

6 hours sleep No sleep

n =14 n = 14

M = 72 M =65

SS = 932 SS = 706

Is there a significant difference between the two conditions? Use a two-tailed test with α = .01.

Remember to use the attached t-distribution chart when obtaining your critical region.

Please answer the question using all of the steps: null in word, alternative in words, null in symbols, alternative in symbols, critical region t, df, all steps in the analysis computing your independent computed t, make a decision, and give a conclusion.

Expert Solution

Step 1

Given:

Sample size of people who get 6 hours of sleep is 14, that is, $n_{1}$ = 14.

Sample mean for the people who get 6 hours of sleep is 72, that is, $M_{1}$ = 72.

Sum of squares for the people who get 6 hours of sleep is 932, that is, $S S_{1}$ =932.

Sample size of people who get no sleep is 14, that is, $n_{2}$ = 14.

Sample mean for the people who get no sleep is 72, that is, $M_{2}$ = 65.

Sum of squares for the people who get no sleep is 932, that is, $S S_{2}$ =706.

Step 2

Solution:

Sample variance for the people who get 6 hours of sleep is:

$\begin{array}{rcl} S_{1}^{2} & = & \frac{S S_{1}}{n_{1} - 1} \\ = & \frac{932}{14 - 1} \\ = & 71.6923 \end{array}$

Sample variance for the people who get no sleep is:

$\begin{array}{rcl} S_{2}^{2} & = & \frac{S S_{2}}{n_{2} - 1} \\ = & \frac{706}{14 - 1} \\ = & 54.3077 \end{array}$

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