Jumbo eggs in Australia, on average, are supposed to weigh $68 \mathrm{~g}$. Tala is in charge of a quality control test that involves weighing a sample of eggs to test $H_{0}: \mu=68 \mathrm{~g}$ versus $H_{\mathrm{a}}: \mu
eq 68 \mathrm{~g}$, where $\mu$ is the mean weight of the eggs in a batch.

Tala sampled 12 eggs from a batch and found a sample mean weight of $68.5 \mathrm{~g}$ and a standard deviation of $1 \mathrm{~g}$. She calculated a test statistic of $t \approx 1.73$ and an approximate $P$-value of 0.111 . Assume that the conditions for inference were met.

Is there sufficient evidence at the $\alpha=0.10$ level to conclude that the mean weight of the eggs in this batch is not equal to $68 \mathrm{~g}$ ?

Choose 1 answer:
(A) Yes, because $1.730.111$.
(B) No, because $1.730.111$.
(c) Yes, because $0.1110.10$.
(D) No, because $0.1110.10$.

Question

Jumbo eggs in Australia, on average, are supposed to weigh $68 \mathrm{~g}$. Tala is in charge of a quality control test that involves weighing a sample of eggs to test $H_{0}: \mu=68 \mathrm{~g}$ versus $H_{\mathrm{a}}: \mu 
eq 68 \mathrm{~g}$, where $\mu$ is the mean weight of the eggs in a batch.

Tala sampled 12 eggs from a batch and found a sample mean weight of $68.5 \mathrm{~g}$ and a standard deviation of $1 \mathrm{~g}$. She calculated a test statistic of $t \approx 1.73$ and an approximate $P$-value of 0.111 . Assume that the conditions for inference were met.

Is there sufficient evidence at the $\alpha=0.10$ level to conclude that the mean weight of the eggs in this batch is not equal to $68 \mathrm{~g}$ ?

Choose 1 answer:
(A) Yes, because $1.73>0.111$.
(B) No, because $1.73>0.111$.
(c) Yes, because $0.111>0.10$.
(D) No, because $0.111>0.10$.

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Identify the null hypothesis $ H_0 $ and the alternative hypothesis $ H_a $: $$ H_0: \mu = 68 \, \text{g} $$ $$ H_a: \mu \neq 68 \, \text{g} $$ ‖ ‖ step 2 ⋮ Given the sample mean $\bar{x} = 68.5 \, \text{g}$, sample standard deviation $s = 1 \, \text{g}$, and sample size $n = 12$, the test statistic $ t $ is calculated as: $$ t = \frac{\bar{x} - \mu}{s / \sqrt{n}} \approx 1.73 $$ ‖ ‖ step 3 ⋮ The p-value associated with the test statistic $ t \approx 1.73 $ is approximately 0.111. ‖ ‖ step 4 ⋮ Compare the p-value to the significance level $\alpha = 0.10$: $$ \text{p-value} = 0.111 $$ $$ \alpha = 0.10 $$ Since $ 0.111 > 0.10 $, we fail to reject the null hypothesis $ H_0 $. ‖ ∻Answer∻ ***D*** ∻Key Concept∻ ⚹p-value comparison with significance level⚹ ∻Explanation∻ ⚹If the p-value is greater than the significance level $\alpha$, we fail to reject the null hypothesis.⚹◊What is the critical value of t for a two-tailed test at $\alpha = 0.10$ with 10 degrees of freedom?⍭ Generate me a similar question◊

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