A study by Conswumer Reports showed that 64 % of supermarket...

A study by Conswumer Reports showed that $64 \%$ of supermarket shoppers believe supermarket brands to be as good as national name brands. To investigate whether this result applies to its own product, the manufacturer of a national name-brand ketchup asked a sample of shoppers whether they believed that supermarket ketchup was as good as the national brand ketchup. a. Formulate the bypotheses that could be used to determine whether the percentage of supermarket shoppers who believe that the supermarket ketchup was as good as the national brand ketchup differed from $64 \%$. b. If a sample of 100 shoppers showed 52 stating that the supermarket brand was as good as the national brand, what is the $p$-value? c. At $a=05$, what is your conclusion? d. Should the narional brand ketchup manufacturer be pleased with this conclusion? Explain.

A study by Conswumer Reports showed that $64 \%$ of supermarket shoppers believe supermarket brands to be as good as national name brands. To investigate whether this result applies to its own product, the manufacturer of a national name-brand ketchup asked a sample of shoppers whether they believed that supermarket ketchup was as good as the national brand ketchup.
a. Formulate the bypotheses that could be used to determine whether the percentage of supermarket shoppers who believe that the supermarket ketchup was as good as the national brand ketchup differed from $64 \%$.
b. If a sample of 100 shoppers showed 52 stating that the supermarket brand was as good as the national brand, what is the $p$-value?
c. At $a=05$, what is your conclusion?
d. Should the narional brand ketchup manufacturer be pleased with this conclusion? Explain.

Key Concepts

Hypothesis Formulation

This concept involves setting up the null hypothesis (H0) and the alternative hypothesis (H1) in statistical tests. For proportion testing, the null hypothesis typically states that the true proportion is equal to a specific value (in this case, 64%), while the alternative hypothesis posits that the true proportion is different from that value. This is essential for determining whether observed sample data provides enough evidence to suggest a deviation from the expected proportion.

p-value Computation

The p-value is the probability of obtaining an effect at least as extreme as the one in your sample data, assuming that the null hypothesis is true. In the context of testing a proportion, the observed number of successes (or responses) is compared to the expected distribution under H0, often using a normal approximation (z-test) when the sample size is large. Calculating the p-value helps to decide whether to reject the null hypothesis.

Significance Level and Decision Making

This concept refers to setting a predetermined threshold (alpha, often 0.05) to decide whether the observed p-value provides sufficient evidence to reject the null hypothesis. The p-value is compared to the alpha level; if the p-value is lower, the result is statistically significant, leading to the rejection of H0, whereas a p-value higher than alpha means that there is insufficient evidence to reject H0. This decision-making framework is central to hypothesis testing.

Practical Significance for Decision Making

Apart from the statistical outcome, practical significance considers the real-world implications of the test results. In scenarios where the finding affects business strategy, like in marketing or product comparison, it's important to assess whether the statistical difference is large enough to have a meaningful impact. Decision makers rely on both the statistical result and its practical relevance to guide actions such as product promotion or reconsidering marketing strategies.

Transcript

00:01 So in this problem, we are interested in seeing whether the supermarket brand of ketchup is as good as the national brand of ketchup.

00:09 So our alternative hypothesis is that mu is not equal to 0 .64, the proportion of people that are interested in the national brand of ketchup.

00:26 And the null hypothesis is that our mu is equal to 0 .64.

00:33 So in order to compute a p value, we need a test statistic.

00:37 And for our test statistic, we need to find a sample proportion.

00:42 Our sample proportion is equal to 52 over 100 because this is the proportion of shoppers from our sample that say the supermarket brand was as good as a national brand, which is equal to 0 .52.

00:55 And our n is equal to 100.

00:58 So in order to find a test statistic, we will need, or p value, we will need to compute a z test test.

01:03 Statistic and this is the formula it is equal to the sample proportion of 0 .52 minus the population proportion of 0 .64 over the square root of the population proportion 0 .64 times 1 minus the population proportion of 0 .64 over the sample size of 100 all square rooted and we get a z value of negative 2 .5.

01:39 So if we plot this on a normal distribution curve, we get z equals 0 here.

01:48 Z equals negative 2 .5 is to the left of z equals 0 and we are interested in finding the area under the curve into the left of z equals negative 2 .5.

01:58 That area represents probability that z is less than or equal to negative 2 .5.

02:05 And in order to compute this value, we need to go to our z table.

02:13 And i already marked where our proportion value is.

02:18 It's 0 .0062.

02:21 But this is not our p value...

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