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Null hypothesis: The population standard deviation of TV watching times for teenagers is equal to or greater than 2.71.
Alternative hypothesis: The population standard deviation of TV watching times for teenagers is less than 2.71.
To find the critical value, we need to look up the appropriate value from the t-distribution table. Since the sample size is small (n=36) and we don't know the population standard deviation, we will use a t-distribution. With a significance level of 0.05 and a one-tailed test (since we're testing if the population standard deviation is less than 2.71), we will find the critical value with 35 degrees of freedom.
The critical value for a t-distribution with 35 degrees of freedom and a significance level of 0.05 is approximately -1.690 (rounding to 3 decimal places).
Next, we need to calculate the test statistic. The formula for the test statistic in this case is:
Test statistic = (sample standard deviation - hypothesized population standard deviation) / (sample standard deviation / sqrt(sample size))
Plugging in the values, we get:
Test statistic = (2.3 - 2.71) / (2.3 / sqrt(36))
= -0.41 / (2.3 / 6)
= -0.41 / 0.383
= -1.07 (rounded to 2 decimal places)
Now, we compare the test statistic to the critical value. Since the test statistic (-1.07) is greater than the critical value (-1.690), we do not have enough evidence to reject the null hypothesis.
In conclusion, based on the given data and significance level of 0.05, we do not have enough evidence to conclude that the population standard deviation of TV watching times for teenagers is less than 2.71.