Math / Data & Statistics

QuestionA $95 \%$ confidence interval for $\mu$ is computed to be $(1.60,4.15)$ . For each of the following hypotheses, state whether $H_{0}$ will be rejected at 0.05 level. Part: $0 / 4$
Part 1 of 4 $H_{0}: \mu=4$ versus $H_{1}: \mu \neq 4$ Since the $95 \%$ confidence interval (Choose one) $\boldsymbol{\nabla} \mu_{0}$ , then $H_{0}$ (Choose one) $\boldsymbol{\nabla}$ be rejected at the 0.05 level.

Studdy Solution

STEP 1

1. The confidence interval for $\mu$ is $(1.60, 4.15)$ .
2. The null hypothesis $H_0$ is $\mu = 4$ .
3. The alternative hypothesis $H_1$ is $\mu \neq 4$ .
4. A 95% confidence interval corresponds to a significance level of $\alpha = 0.05$ .

STEP 2

1. Determine if the null hypothesis value $\mu_0 = 4$ is within the confidence interval.
2. Decide whether to reject $H_0$ based on the confidence interval.

STEP 3

Identify the confidence interval:
$(1.60, 4.15)$

STEP 4

Check if $\mu_0 = 4$ is within the interval $(1.60, 4.15)$ .
Since $4$ is within the interval $(1.60, 4.15)$ , $\mu_0$ is included.

STEP 5

Since $\mu_0 = 4$ is within the confidence interval, we do not reject $H_0$ .
Conclusion:
Since the $95\%$ confidence interval includes $\mu_0 = 4$ , then $H_0$ will not be rejected at the 0.05 level.

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