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Answer :
To solve this problem, follow these steps:
1. Since we are testing a population mean and the population standard deviation is unknown, we use the [tex]$t$[/tex]-distribution.
2. The sample size is [tex]$n = 35$[/tex], so the degrees of freedom (df) is calculated as:
[tex]$$
\text{df} = n - 1 = 35 - 1 = 34.
$$[/tex]
3. For a left-tailed test with a significance level of [tex]$\alpha = 0.05$[/tex], we need to find the [tex]$t$[/tex]-value such that the cumulative probability to the left of this value is [tex]$0.05$[/tex]. This means we are looking for the quantile [tex]$t_{0.05,34}$[/tex].
4. Consulting a [tex]$t$[/tex]-distribution table or using statistical software gives:
[tex]$$
t_{0.05,34} \approx -1.691.
$$[/tex]
5. Rounding to three decimal places, the critical value is [tex]$-1.691$[/tex].
Thus, the correct answer is:
B. [tex]$-1.691$[/tex]
1. Since we are testing a population mean and the population standard deviation is unknown, we use the [tex]$t$[/tex]-distribution.
2. The sample size is [tex]$n = 35$[/tex], so the degrees of freedom (df) is calculated as:
[tex]$$
\text{df} = n - 1 = 35 - 1 = 34.
$$[/tex]
3. For a left-tailed test with a significance level of [tex]$\alpha = 0.05$[/tex], we need to find the [tex]$t$[/tex]-value such that the cumulative probability to the left of this value is [tex]$0.05$[/tex]. This means we are looking for the quantile [tex]$t_{0.05,34}$[/tex].
4. Consulting a [tex]$t$[/tex]-distribution table or using statistical software gives:
[tex]$$
t_{0.05,34} \approx -1.691.
$$[/tex]
5. Rounding to three decimal places, the critical value is [tex]$-1.691$[/tex].
Thus, the correct answer is:
B. [tex]$-1.691$[/tex]
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