A study of young children was designed to increase their intake of whole-grain, rather than regular-grain, snacks. At the end of the study, the 79 children who participated in the study were presented with a choice between a regular-grain snack and a whole-grain alternative. The whole-grain alternative was chosen by 55 children. You want to examine the possibility that the children are equally likely to choose each type of snack. H_0: p = 0.5 H_a: p != 0.5 Perform the significance test. (Use alpha = 0.01. Round your test statistic to two decimal places and your P-value to four decimal places.)

A study of young children was designed to increase their intake of whole-grain, rather than regular-grain, snacks. At the end of the study, the 79 children who participated in the study were presented with a choice between a regular-grain snack and a whole-grain alternative. The whole-grain alternative was chosen by 55 children. You want to examine the possibility that the children are equally likely to choose each type of snack. H_0: p = 0.5 H_a: p != 0.5 Perform the significance test. (Use alpha = 0.01. Round your test statistic to two decimal places and your P-value to four decimal places.)

Question
Study design
asked 2021-02-24
A study of young children was designed to increase their intake of whole-grain, rather than regular-grain, snacks. At the end of the study, the 79 children who participated in the study were presented with a choice between a regular-grain snack and a whole-grain alternative. The whole-grain alternative was chosen by 55 children. You want to examine the possibility that the children are equally likely to choose each type of snack.
\(\displaystyle{H}_{{0}}:{p}={0.5}\)
\(\displaystyle{H}_{{a}}:{p}\ne{0.5}\)
Perform the significance test. (Use \(\displaystyle\alpha={0.01}\). Round your test statistic to two decimal places and your P-value to four decimal places.)

Answers (1)

2021-02-25

Step 1
Given Information
No of children participated (n) = 79
No of children chosen (x) = 55
\(\displaystyle\hat{{{p}}}=\frac{{x}}{{n}}=\frac{{55}}{{79}}={0.696}\)
Null and alternative Hypothesis
Null Hypothesis: p = 0.5
Alternative Hypothesis: \(\displaystyle{p}\ne{0.5}\)
Test statistic
\(\displaystyle{z}=\frac{{\hat{{{p}}}-{p}}}{{\sqrt{{\frac{{{p}{\left({1}-{p}\right)}}}{{n}}}}=\frac{{{0.696}-{0.5}}}{{\sqrt{{\frac{{{0.5}\times{\left({1}-{0.5}\right)}}}{{79}}}}}}={3.48}}}\)
Step 2
p-value
\(\displaystyle{p}-{v}{a}{l}{u}{e}={2}{x}{P}{\left({Z}{>}{3.48}\right)}={2}{x}{\left({1}–{P}{\left({Z}{<}{3.48}\right)}\right)}={2}{x}{\left({1}–{0.999748}\right)}={0.000501}\)
\(P(Z<3.48)=0.999748\ \text{(Using Excel function=NORM.S.DIST(B 20, TRUE)}\)
p-value < 0.01,
So, reject null hypothesis and conclude that children are not equally likely to choose each type of snack.

0

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The table below shows the number of people for three different race groups who were shot by police that were either armed or unarmed. These values are very close to the exact numbers. They have been changed slightly for each student to get a unique problem.
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White - 1176
Hispanic - 378
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Suspect was unarmed:
Black - 60
White - 67
Hispanic - 38
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Total:
Black - 603
White - 1243
Hispanic - 416
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Give your answer as a decimal to at least three decimal places.
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This just means that the percentage of times that both things happen equals the individual percentages multiplied together (Only if they are Independent of each other).
Therefore, if a person's race is independent of whether they were killed being unarmed then the percentage of black people that are killed while being unarmed should equal the percentage of blacks times the percentage of Unarmed. Let's check this. Multiply your answer to part a (percentage of blacks) by your answer to part b (percentage of unarmed).
Remember, the previous answer is only correct if the variables are Independent.
d) Now let's get the real percent that are Black and Unarmed by using the table?
If answer c is "significantly different" than answer d, then that means that there could be a different percentage of unarmed people being shot based on race. We will check this out later in the course.
Let's compare the percentage of unarmed shot for each race.
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f) What percent are Hispanic and Unarmed?
If you compare answers d, e and f it shows the highest percentage of unarmed people being shot is most likely white.
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This is because there are more white people in the United States than any other race and therefore there are likely to be more white people in the table. Since there are more white people in the table, there most likely would be more white and unarmed people shot by police than any other race. This pulls the percentage of white and unarmed up. In addition, there most likely would be more white and armed shot by police. All the percentages for white people would be higher, because there are more white people. For example, the table contains very few Hispanic people, and the percentage of people in the table that were Hispanic and unarmed is the lowest percentage.
Think of it this way. If you went to a college that was 90% female and 10% male, then females would most likely have the highest percentage of A grades. They would also most likely have the highest percentage of B, C, D and F grades
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