The role of GABA or gamma-aminobutyric acid is involved in down-regulating the activity of neurons in the human brain and is linked to mental health disorders as: anxiety, depression, autism, schizophrenia, bipolarity, and epilepsy among many others. You sample 50 males without any previous diagnosis of a mental disorder between the ages of 25-31 at a school and report that the average amount of GABA in the sample is 1.16 umol/cm3. Likewise you also report that the standard deviation of the sample is .09 umol/cm3. Construct a 98.64% confidence interval using the sample standard deviation as an estimate for the population standard deviation. Report your upper value as the answer to this question to two decimal places and appropriate rounding.
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To construct a confidence interval for the mean, you can use the formula:
Confidence Interval = sample mean ± (critical value * standard deviation / square root of sample size)
Step 1: Find the critical value
Since the sample size is large (n = 50), we will use the z-distribution and a confidence level of 98.64%. To find the critical value, we need to determine the z-score associated with this confidence level.
The z-score can be found using a standard normal distribution table or a statistical calculator. For a two-tailed 98.64% confidence level, we divide the remaining 1 - 0.9864 by 2 to get the tail probability of 0.0068.
Using a standard normal distribution table or calculator, we find the z-score for the tail probability of 0.0068 is approximately 2.61.
Step 2: Calculate the confidence interval
Now, we can calculate the confidence interval using the formula mentioned earlier:
Confidence Interval = sample mean ± (critical value * standard deviation / square root of sample size)
Given:
Sample mean (x̄) = 1.16 umol/cm3
Standard deviation (s) = 0.09 umol/cm3
Sample size (n) = 50
Critical value (z) = 2.61
Confidence Interval = 1.16 ± (2.61 * 0.09 / √50)
Step 3: Solve the expression
Confidence Interval = 1.16 ± (2.61 * 0.09 / 7.07)
Confidence Interval = 1.16 ± 0.033
Finally, calculate the upper value of the confidence interval:
Upper Value = 1.16 + 0.033
Upper Value = 1.193
Therefore, the upper value of the 98.64% confidence interval is 1.193 umol/cm3.