Questions: At "Stavros Restaurant" the owner is told by his chef that 25% of the customers order vegetarian food. The owner wants to check the validity of the chef's assertion so he checks a random sample of 20 orders, only to find 2 vegetarian orders. a) Is there evidence, at the 10% level of significance, that the proportion of vegetarian orders is lower than 25%? At "Mavros Restaurant" the owner is told by his waiters that 25% of the customers order vegetarian food. The owner wants to check the validity of the waiters' belief so he checks a random sample of 100 orders. b) Given that there are 31 vegetarian orders in the sample, use a distributional approximation, to test at the 5% level of significance, the belief of the waiters at "Mavros Restaurant". No credit will be given in part b) to answers which do not use a distributional approximation.

At "Stavros Restaurant" the owner is told by his chef that 25% of the customers order vegetarian food. The owner wants to check the validity of the chef's assertion so he checks a random sample of 20 orders, only to find 2 vegetarian orders.
a) Is there evidence, at the 10% level of significance, that the proportion of vegetarian orders is lower than 25%?

At "Mavros Restaurant" the owner is told by his waiters that 25% of the customers order vegetarian food. The owner wants to check the validity of the waiters' belief so he checks a random sample of 100 orders.
b) Given that there are 31 vegetarian orders in the sample, use a distributional approximation, to test at the 5% level of significance, the belief of the waiters at "Mavros Restaurant".

No credit will be given in part b) to answers which do not use a distributional approximation.
Transcript text: At "Stavros Restaurant" the owner is told by his chef that $25 \%$ of the customers order vegetarian food. The owner wants to check the validity of the chef's assertion so he checks a random sample of 20 orders, only to find 2 vegetarian orders. a) Is there evidence, at the $10 \%$ level of significance, that the proportion of vegetarian orders is lower than $25 \%$ ? At "Mavros Restaurant" the owner is told by his waiters that $25 \%$ of the customers order vegetarian food. The owner wants to check the validity of the waiters' belief so he checks a random sample of 100 orders. b) Given that there are 31 vegetarian orders in the sample, use a distributional approximation, to test at the $5 \%$ level of significance, the belief of the waiters at "Mavros Restaurant". No credit will be given in part b) to answers which do not use a distributional approximation.
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Solution

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Solution Steps

Step 1: Hypothesis Testing at Stavros Restaurant

The owner of Stavros Restaurant wants to test the chef's assertion that 25%25\% of the customers order vegetarian food. We set up the following hypotheses:

  • Null Hypothesis (H0H_0): p=0.25p = 0.25
  • Alternative Hypothesis (HaH_a): p<0.25p < 0.25

Using a sample of n=20n = 20 orders, with k=2k = 2 vegetarian orders, we calculate the sample proportion:

p^=kn=220=0.1 \hat{p} = \frac{k}{n} = \frac{2}{20} = 0.1

The test statistic is calculated as:

Z=p^p0p0(1p0)n=0.10.250.25(10.25)20=1.5492 Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} = \frac{0.1 - 0.25}{\sqrt{\frac{0.25(1 - 0.25)}{20}}} = -1.5492

The corresponding p-value is 0.06070.0607.

At the 10%10\% significance level, the critical region is defined as Z<1.2816Z < -1.2816. Since 1.5492<1.2816-1.5492 < -1.2816, we reject the null hypothesis.

Step 2: Hypothesis Testing at Mavros Restaurant

The owner of Mavros Restaurant also wants to test the waiters' belief that 25%25\% of the customers order vegetarian food. The hypotheses are:

  • Null Hypothesis (H0H_0): p=0.25p = 0.25
  • Alternative Hypothesis (HaH_a): p0.25p \neq 0.25

In this case, the sample size is n=100n = 100 with k=31k = 31 vegetarian orders. The sample proportion is:

p^=kn=31100=0.31 \hat{p} = \frac{k}{n} = \frac{31}{100} = 0.31

The test statistic is calculated as:

Z=p^p0p0(1p0)n=0.310.250.25(10.25)100=1.3856 Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} = \frac{0.31 - 0.25}{\sqrt{\frac{0.25(1 - 0.25)}{100}}} = 1.3856

The corresponding p-value is 0.16590.1659.

At the 5%5\% significance level, the critical region is defined as Z<1.96Z < -1.96 or Z>1.96Z > 1.96. Since 1.38561.3856 does not fall into the critical region, we fail to reject the null hypothesis.

Final Answer

For Stavros Restaurant, there is evidence at the 10%10\% level of significance that the proportion of vegetarian orders is lower than 25%25\%. For Mavros Restaurant, there is not enough evidence at the 5%5\% level of significance to reject the waiters' belief.

Stavros: Reject H0, Mavros: Fail to reject H0\boxed{\text{Stavros: Reject } H_0, \text{ Mavros: Fail to reject } H_0}

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