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An educational consulting firm is trying to decide whether high school students who have never before used a handheld calculator can solve a certain type of problem more easily with a calculator that uses reverse Polish logic or one that does not use this logic. A sample of 25 students is selected and allowed to practice on both calculators. Then each student is asked to work one problem on the reverse Polish calculator and a similar problem on the other. Let \(p=P(S)\), where \(S\) indicates that a student worked the problem more quickly using reverse Polish logic than without, and let \(X=\) number of \(S\) 's. a. If \(p=.5\), what is \(P(7 \leq X \leq 18)\) ? b. If \(p=.8\), what is \(P(7 \leq X \leq 18)\) ? c. If the claim that \(p=.5\) is to be rejected when either \(X \leq\) 7 or \(X \geq 18\), what is the probability of rejecting the claim when it is actually correct? d. If the decision to reject the claim \(p=.5\) is made as in part (c), what is the probability that the claim is not rejected when \(p=.6\) ? When \(p=.8\) ? e. What decision rule would you choose for rejecting the claim \(p=.5\) if you wanted the probability in part (c) to be at most .01?

Step by step solution

01

Understanding the Problem

Identify and define the given problem. We have a sample size of 25 students, and each student can solve the problem either faster with reverse Polish logic or without. Let \( p \) be the probability that a student solves the problem faster using reverse Polish logic. \( X \) is the number of students who solve the problem faster using reverse Polish logic. The task is to calculate certain probabilities for different values of \( p \).

02

Determine Distribution Type

Since \( X \) represents the number of successes (students solving the problem faster using reverse Polish logic) in 25 trials, and each trial has two possible outcomes, \( X \) follows a binomial distribution: \( X \sim \text{Binomial}(n=25, p) \).

03

Calculate Probability in (a) with \( p=0.5 \)

To find \( P(7 \leq X \leq 18) \) when \( p=0.5 \), use the binomial distribution formula or a statistical table/computational tool. Calculate the cumulative probabilities from \( 7 \) to \( 18 \) using \( P(X = k) = \binom{25}{k} (0.5)^k (0.5)^{25-k} \) for \( k=7 \) to \( 18 \), and sum them up.

04

Calculate Probability in (b) with \( p=0.8 \)

Follow a similar procedure to step 3, but now use \( p=0.8 \). Calculate \( P(7 \leq X \leq 18) \) by finding individual probabilities \( P(X = k) \) for \( k=7 \) to \( 18 \) using \( P(X = k) = \binom{25}{k} (0.8)^k (0.2)^{25-k} \) and sum them up.

05

Calculate Probability of Rejecting the Claim in (c)

For part (c), determine the probability that \( X \leq 7 \) or \( X \geq 18 \) when \( p=0.5 \). This is equivalent to finding \( P(X \leq 7) + P(X \geq 18) \) using the binomial distribution. You can use cumulative binomial distribution tables or software for precise values.

06

Calculate Non-Rejection Probabilities in (d)

For the decision rule in part (c), compute the non-rejection probability \( P(7 < X < 18) \) when the true \( p=0.6 \) and when \( p=0.8 \). Use the binomial distribution formula to find the cumulative probability between \( 7 \) and \( 18 \) for each \( p \), and subtract from 1 to find the non-rejection probabilities.

07

Determine Decision Rule for (e) with \( P \leq 0.01 \)

Find a decision rule for \( X \leq k \) or \( X \geq m \) such that \( P(\text{rejecting the null hypothesis} | p=0.5) \leq 0.01 \). Experiment with different values of \( k \) and \( m \) to meet the probability constraint using cumulative probabilities from the binomial distribution with \( p=0.5 \), so that the risk of a Type I error (false rejection) is controlled.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Reverse Polish Logic

Reverse Polish Logic (RPL) is a mathematical notation where operators follow their operands. For example, instead of writing \(3 + 4\), it is written as \(3\ 4 +\). This eliminates the need for parentheses employed in infix notation (standard notation), as the order of operations is unambiguously understood. RPL is often used in calculators to process expressions more efficiently and minimize errors in programming.The key advantages of Reverse Polish Logic include:

• Fewer keystrokes needed compared to infix notation.
• Reduced risk of operator precedence mistakes.
• Elimination of the need for parentheses.

In the exercise, students solve problems using calculators with RPL to determine if this method allows for faster problem-solving compared to non-RPL calculators. Understanding RPL can aid in efficient computation and effective time management.

Probability Calculations

Probability calculations are essential to determine the likelihood of a certain event happening. In the context of the exercise, these calculations are based on the binomial distribution, which models the number of successes in a fixed number of independent trials, where each trial has two possible outcomes.For instance, when calculating \( P(7 \leq X \leq 18) \), we're interested in the probability that the number of students who solve the problem faster using Reverse Polish logic is between 7 and 18. The probability mass function for the binomial distribution is given by:\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]where \(n\) is the total number of trials (e.g., 25 students), \(k\) is the number of successes, and \(p\) is the probability of a single success. Using this formula helps calculate probabilities for each outcome to determine cumulative probability.

Statistical Hypothesis Testing

Statistical hypothesis testing involves making a decision about a population based on sample data. It starts with two hypotheses: the null hypothesis (usually a statement of "no effect" or "no difference") and the alternative hypothesis.In the exercise, the null hypothesis is \( p = 0.5 \), suggesting there's no difference in speed. When calculating the probability of rejecting this claim inaccurately (Type I error), hypothesis testing helps determine the significance level often set at 0.05 or 0.01.Steps include:

• Selecting a significance level, \( \alpha \).
• Calculating the test statistic using the sample result.
• Comparing the statistic to a critical value to decide whether to reject the null hypothesis.

This process helps understand whether perceived differences are statistically valid or due to random chance.

Cumulative Probability

Cumulative probability refers to the probability that a random variable is less than or equal to a certain value. It is crucial for evaluating various outcomes collectively. Instead of assessing the probability of each event separately, cumulative probability sums them up until reaching a certain point.In the binomial distribution setup of our exercise, we calculate cumulative probabilities to determine the range \( 7 \leq X \leq 18 \). This involves summing the probabilities from \( P(X=7) \) to \( P(X=18) \), providing insight into how likely a collective set of outcomes is.The cumulative distribution function (CDF) is key:\[ P(a \leq X \leq b) = \sum_{k=a}^{b} P(X=k) \]Cumulative probabilities simplify the analysis of events over a continuous range and are useful in both descriptive statistics and hypothesis testing.