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A physical therapist claims that 75% of people lose their balance if they close their eyes while standing on one foot. A gymnast feels that the rate is less than 75%. To test this, she randomly selects 100 people and finds that 80 of them lose their balance while standing on one foot.

To determine if these data provide convincing evidence that the proportion of people who lose their balance while standing on one foot and closing their eyes is less than 75%, 150 trials of a simulation are conducted. The gymnast is testing the hypotheses:

\[ H_0: p = 75\% \]
\[ H_a: p < 75\% \]

where \( p \) is the true proportion of people who will lose their balance while standing on one foot and closing their eyes.

Based on the results of the simulation, what is the estimate of the P-value of the test?

**A dotplot is provided.**

A number line labeled "simulated sample percentage of people who lose balance" goes from 64 to 88 percent.

- 64: 3
- 66: 3
- 67: 4
- 68: 8
- 69: 5
- 70: 7
- 71: 8
- 72: 11
- 73: 15
- 74: 12
- 75: 11
- 76: 8
- 77: 18
- 78: 10
- 79: 10
- 80: 5
- 81: 4
- 82: 2
- 83: 2
- 84: 1
- 85: 2
- 87: 1

What is the estimate of the P-value of the test?

A. 0.1133
B. 0.17
C. 0.887
D. 0.92

Answer :

Final answer:

The p-value is calculated by focusing on the area to the left of the observed proportion (80%) in the distribution of simulated proportions, counting all the percentages of people who lose balance that are less than 80% and dividing by the total number of simulations. If the p-value is less than a chosen significance level, we reject the null hypothesis and support the alternative hypothesis.

Explanation:

In this problem, the gymnast is using hypothesis testing with a null hypothesis (H0) that the true proportion (p) of people who will lose their balance while standing on one foot and closing their eyes equals 75%, and an alternative hypothesis (Ha) that p is less than 75%.

Based on her sample of 100 people, 80% of people lose their balance. This becomes her observed or experimental proportion (p'), which is higher than the hypothesized 75%. However, in the simulation that she conducted, she gathered data on how varied the outcomes could be if the null hypothesis were true.

The p-value of the test represents the probability of obtaining results as extreme as the observed data given that the null hypothesis is true. To calculate the p-value, we focus on the area to the left of the observed proportion (80%) in the distribution of simulated proportions, counting all the percentages of people who lose balance that are less than 80%. Summing the proportions associated with these percentages and dividing by the total number of simulations (150) gives us the estimate of the p-value.

Note that if the p-value is less than a chosen significance level (usually 0.05), we will reject the null hypothesis (p = 75% in this case) and conclude that the data provides convincing evidence to support the alternative hypothesis (p < 75%).

Learn more about Hypothesis Testing here:

https://brainly.com/question/34171008

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