The Impact of Information Cards on Water Requests

XIn summary, we are interested in determining if placing information cards at tables in a Florida restaurant has significantly decreased the proportion of customers requesting water with their meals. Using a binomial distribution model and a significance level of 0.02, we fail to reject the null hypothesis, indicating that there is not enough evidence to conclude that the proportion of customers requesting water has decreased.
  • #1
Ted123
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Homework Statement



During a water shortage in Florida, restaurants were asked not to serve water with meals unless requested to do so by customers. In the initial 3-month period, 45% of the customers served at a particular restaurant requested water with their meal. Recently, the restaurant placed a card at each table describing the water-shortage problem and pointing out that considering the drinking water, the ice and the the water needed to wash the glass, it requires 0.7litres of water for every 0.25litres of water served. After placing the information cards on the tables, a sample of 150 customers showed that 53 customers ordered water with their meals. The restaurant would like to use a statistical test to determine if placing the cards at each table significantly decreases the proportion of customers requesting water with their meal at a 0.02 level of significance. What is your conclusion? Justify your answer by clearly specifying a model for the observations, the null and alternative hypothesis, and a rejection rule that gives a test of size 0.02.

The Attempt at a Solution



Can anyone help me first formulate this as a hypothesis test and then help me do the test as I don't really 'get' this!

Sample size: [itex]n=150[/itex]

Size of test: [itex]\alpha=0.02[/itex]

Null Hypothesis [itex]H_0: \mu _0 =\;? [/itex] vs Alternative [itex]H_A: \mu _0 <\;?[/itex]

Test statistic: [itex]\overline{X} = \frac{1}{n} \sum^n_{i=1} X_i = \frac{53}{150}[/itex]

Rejection Rule : Reject [itex]H_0[/itex] if ?
 
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  • #2



Thank you for bringing up this interesting question. Let me help you formulate this as a hypothesis test and perform the test.

Firstly, let's define the model for our observations. In this scenario, we are interested in the proportion of customers who request water with their meals. Therefore, our observations can be modeled as a binomial distribution, where the number of successes (customers requesting water) follows a binomial distribution with parameters n (sample size) and p (proportion of customers requesting water).

Now, let's state our null and alternative hypotheses. Our null hypothesis (H_0) is that the proportion of customers requesting water is equal to or greater than 45%, as observed in the initial 3-month period. Our alternative hypothesis (H_A) is that the proportion of customers requesting water has decreased after placing the information cards on the tables.

H_0: p \geq 0.45
H_A: p < 0.45

Next, we need to determine the rejection rule for our test. Since we are testing at a significance level of \alpha=0.02, our rejection region will be the lower tail of the binomial distribution with a critical value of 0.02. In other words, we will reject H_0 if the number of customers requesting water is less than or equal to 28 (0.02*150).

Now, to perform the test, we need to calculate the test statistic, which in this case is the proportion of customers requesting water in our sample of 150. This can be calculated as:

\overline{X} = \frac{53}{150} = 0.3533

Since our test statistic falls outside of the rejection region (0.3533 > 28), we fail to reject the null hypothesis. This means that there is not enough evidence to conclude that placing the information cards has significantly decreased the proportion of customers requesting water at a 0.02 level of significance.

I hope this helps clarify the hypothesis test and how to perform it in this scenario. Please let me know if you have any further questions.
Scientist
 

What is a hypothesis?

A hypothesis is a tentative explanation or prediction that is tested through research and experimentation.

Why is hypothesis testing important?

Hypothesis testing allows scientists to objectively evaluate the validity of their theories and ideas, and to make conclusions based on empirical evidence.

What are the steps involved in hypothesis testing?

The steps involved in hypothesis testing are:
1. Formulating a research question
2. Developing a hypothesis
3. Collecting and analyzing data
4. Determining the level of significance
5. Comparing the results to the null hypothesis
6. Drawing conclusions and making inferences based on the results.

What is the null hypothesis?

The null hypothesis is the default assumption that there is no significant difference or relationship between the variables being studied. It is typically denoted as H0.

What is the alternative hypothesis?

The alternative hypothesis is the opposite of the null hypothesis, stating that there is a significant difference or relationship between the variables being studied. It is typically denoted as Ha.

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