answer:To determine if the manufacturer's claim can be rejected, we will perform a hypothesis test. The null hypothesis (H0) is that the mean weight is equal to the claimed value: H0: μ = 500 grams The alternative hypothesis (H1) is that the mean weight is less than 500 grams: H1: μ < 500 grams We will use a one-sample t-test for this hypothesis test since we have a single sample and want to compare the mean to a known population value. The test statistic is calculated as follows: t = (sample mean - population mean) / (standard error) The standard error is calculated as: standard error = (standard deviation) / sqrt(sample size) In this case, the standard deviation is 10 grams, the sample size is 25, and the sample mean is 490 grams. Plugging these values into the formula for the standard error, we get: standard error = 10 grams / sqrt(25) = 2 grams Now we can calculate the test statistic: t = (490 grams - 500 grams) / 2 grams = -5 grams / 2 grams = -2.5 Next, we need to find the critical value for the t-distribution with 24 degrees of freedom (one less than the sample size) and a significance level of 0.05 for a left-tailed test (since H1 states that the mean weight is less than 500 grams). Using a t-table or a calculator, we find the critical value to be approximately -1.711. Since our calculated test statistic (-2.5) is less than the critical value (-1.711), we reject the null hypothesis. At a significance level of 0.05, there is significant evidence to conclude that the mean weight of the cereal boxes is less than the manufacturer's claimed value of 500 grams.,To determine if the manufacturer's claim can be rejected at a significance level of 0.05, we will perform a one-sample t-test. Step 1: State the null and alternative hypotheses Null hypothesis (H0): μ = 500 grams (The mean weight of the cereal box is 500 grams) Alternative hypothesis (H1): μ ≠ 500 grams (The mean weight of the cereal box is not 500 grams) Step 2: Calculate the test statistic The test statistic for a one-sample t-test is given by the formula: t = (x̄ - μ) / (s / √n) where x̄ is the sample mean, μ is the population mean, s is the sample standard deviation, and n is the sample size. In this case, x̄ = 490 grams, μ = 500 grams, s = 10 grams, and n = 25. t = (490 - 500) / (10 / √25) = -10 / (10 / 5) = -10 / 2 = -5 Step 3: Determine the critical value Since we are performing a two-tailed test at a significance level of 0.05, we need to find the critical value for a t-distribution with 24 degrees of freedom (n - 1 = 25 - 1 = 24). Using a t-table or calculator, we find the critical value to be approximately ±2.064. Step 4: Compare the test statistic to the critical value Our test statistic, t = -5, is more extreme than the critical value of ±2.064. Step 5: Make a decision Since the test statistic is more extreme than the critical value, we reject the null hypothesis. Therefore, we can conclude that the manufacturer's claim that the mean weight of the cereal box is 500 grams can be rejected at a significance level of 0.05.,To determine if the manufacturer's claim can be rejected at a significance level of 0.05, we need to perform a hypothesis test using the given information. Null hypothesis (H0): μ = 500 grams Alternative hypothesis (H1): μ ≠ 500 grams We will use the following information for the test: - Sample size (n) = 25 - Sample mean (x̄) = 490 grams - Population standard deviation (σ) = 10 grams - Significance level (α) = 0.05 We will use the t-test for this scenario since the population standard deviation is known. Calculate the t-score using the formula: t = (x̄ - μ) / (σ / √n) t = (490 - 500) / (10 / √25) t = -10 / (10 / 5) t = -10 / 2 t = -5 Now, we need to find the critical t-value(s) at a significance level of 0.05. Since this is a two-tailed test, we'll use the t-dist table for degrees of freedom (df) = n - 1 = 25-1 = 24. The critical t-value(s) at 0.05 level of significance are approximately -2.064 and 2.064. The calculated t-score (-5) falls outside the range of the critical t-values (-2.064, 2.064). Therefore, we can reject the null hypothesis (H0) and conclude that there is sufficient evidence to reject the manufacturer's claim that the mean weight of the cereal box is 500 grams at a significance level of 0.05.

answer:To find the probability of flipping a fair coin seven times and getting exactly four heads and three tails, we will use the binomial probability formula: P(x) = (n! / (k!(n-k)!)) * (p^k) * (q^(n-k)) where: - P(x) is the probability of getting exactly k successes (in this case, heads) in n trials (flips) - n is the total number of flips (7 in this case) - k is the number of successful outcomes (4 heads) - p is the probability of success (0.5 for a fair coin) - q is the probability of failure (1 - p, or 0.5 for a fair coin) Plugging in the values, we get: P(4) = (7! / (4!(7-4)!)) * (0.5^4) * (0.5^(7-4)) Calculating the factorials: 7! = 7 * 6 * 5 * 4 * 3 * 2 * 1 = 5040 4! = 4 * 3 * 2 * 1 = 24 3! = 3 * 2 * 1 = 6 1! = 1 Simplifying the expression: P(4) = (5040 / (24 * 6)) * (0.0625) * (0.5^3) P(4) = (20) * (0.0625) * (0.125) P(4) = 0.15625 So the probability of flipping the coin seven times and getting exactly four heads and three tails is 0.15625 or 15.625%.,To find the probability of getting exactly 4 heads and 3 tails in 7 coin flips, we can use the binomial probability formula: P(X = k) = (nCk) * (p^k) * (q^(n-k)) where: - n is the total number of flips (7 in this case) - k is the number of successful outcomes (4 heads) - p is the probability of a successful outcome (0.5 for heads) - q is the probability of a failure outcome (0.5 for tails) - nCk is the number of combinations of n items taken k at a time, which can be calculated as n! / (k!(n-k)!) Plugging in the values: P(X = 4) = (7C4) * (0.5^4) * (0.5^3) First, we calculate 7C4: 7! / (4!(7-4)!) = 7! / (4!3!) = (7*6*5) / (4*3*2) = 35 Now, we calculate the probabilities: (0.5^4) = 0.0625 (0.5^3) = 0.125 Finally, we put it all together: P(X = 4) = 35 * 0.0625 * 0.125 = 0.2734375 So, the probability of flipping the coin seven times and getting exactly four heads and three tails is approximately 0.2734 or 27.34%., To calculate the probability, we need to determine the number of ways to get four heads and three tails when flipping a coin seven times, then divide that by the total number of possible outcomes when flipping a coin seven times. First, let's find the number of ways to choose which flips will be heads. We want to choose 4 positions out of 7 for the heads: Ways to choose heads = C(7, 4) = 7! / (4! * (7 - 4)!) = 7! / (4! * 3!) = (7 * 6 * 5) / (3 * 2 * 1) = 35 Now, let's find the total number of possible outcomes when flipping a coin seven times. Since a coin has two possible outcomes (heads or tails), and we're flipping it seven times: Total outcomes = 2^7 = 128 Finally, we can calculate the probability by dividing the number of ways to get four heads and three tails by the total number of possible outcomes: Probability = Ways to choose heads / Total outcomes = 35 / 128 ≈ 0.2734 So, the probability of flipping a fair coin seven times and getting exactly four heads and three tails is approximately 0.2734 or 27.34%.

answer:To calculate the probability of getting exactly 4 heads in 10 coin flips, we can use the binomial probability formula: P(X = k) = (nCk) * (p^k) * (q^(n-k)) where: - P(X = k) is the probability of getting exactly k successes (in this case, heads) in n trials (coin flips) - nCk (read as "n choose k") is the number of combinations of choosing k successes from n trials, which can be calculated as n! / (k!(n-k)!) - p is the probability of success (getting heads) in a single trial (coin flip), which is 0.5 for a fair coin - q is the probability of failure (getting tails) in a single trial, which is also 0.5 for a fair coin - n is the number of trials (coin flips), which is 10 in this case - k is the number of successes (heads) we want, which is 4 in this case Plugging in the values: P(X = 4) = (10C4) * (0.5^4) * (0.5^(10-4)) First, let's calculate 10C4: 10C4 = 10! / (4!(10-4)!) = 10! / (4!6!) = (10*9*8*7) / (4*3*2*1) = 210 Now, let's calculate the rest of the formula: P(X = 4) = 210 * (0.5^4) * (0.5^6) = 210 * 0.0625 * 0.015625 = 210 * 0.0009765625 ≈ 0.205078125 So, the probability of getting exactly 4 heads in 10 coin flips is approximately 0.2051, or 20.51%., To calculate the probability, we need to determine the number of successful outcomes (landing on heads 4 times) and divide it by the total possible outcomes (all possible combinations of heads and tails in 10 flips). First, let's find the number of successful outcomes. We want to choose 4 positions out of 10 to be heads. This can be calculated using combinations: C(10, 4) = 10! / (4! * (10 - 4)!) = 10! / (4! * 6!) = 210 So, there are 210 successful outcomes where the coin lands on heads exactly 4 times. Next, let's find the total possible outcomes. Since a coin flip has 2 possibilities (heads or tails), a sequence of 10 flips will have: 2^10 = 1024 possible outcomes. Finally, we can calculate the probability: Probability = Successful outcomes / Total possible outcomes = 210 / 1024 ≈ 0.2051 So, the probability that the coin lands on heads exactly 4 times in 10 flips is approximately 0.2051 or 20.51%.,To find the probability of getting exactly 4 heads in 10 flips, we will use the binomial probability formula: P(X=k) = (nCk) * (p^k) * (q^(n-k)) Where: - P(X=k) is the probability of getting exactly k successes (in this case, heads) in n trials (flips) - nCk (read "n choose k") is the number of combinations of choosing k successes from n trials, calculated as n! / (k!(n-k)!) - p is the probability of success (getting heads) in a single trial, which is 0.5 for a fair coin - q is the probability of failure (getting tails) in a single trial, which is also 0.5 for a fair coin - n is the number of trials (flips), which is 10 in this case - k is the number of successes (heads) we want, which is 4 in this case Plugging in the values, we get: P(X=4) = (10C4) * (0.5^4) * (0.5^(10-4)) First, we calculate 10C4: 10C4 = 10! / (4!(10-4)!) = 210 Now, we calculate (0.5^4) and (0.5^(10-4)): 0.5^4 = 0.0625 0.5^6 = 0.015625 Finally, we multiply the results: P(X=4) = 210 * 0.0625 * 0.015625 = 0.205078125 So, the probability of getting exactly 4 heads in 10 flips of a fair coin is approximately 0.2051 or 20.51%.

answer:To find the probability of getting exactly 50 heads in 100 flips of a fair coin, we can use the binomial probability formula: P(X = k) = (nCk) * (p^k) * (q^(n-k)) where: - n is the number of trials (flips), in this case, 100 - k is the number of successful outcomes (heads), in this case, 50 - p is the probability of success (getting heads) in a single trial, which is 0.5 for a fair coin - q is the probability of failure (getting tails) in a single trial, which is also 0.5 for a fair coin - nCk is the number of combinations of choosing k successes from n trials, calculated as n! / (k!(n-k)!) Plugging in the values: P(X = 50) = (100C50) * (0.5^50) * (0.5^50) First, we calculate 100C50: 100! / (50! * 50!) = 100891344545564193334812497256 Now, we calculate (0.5^50) * (0.5^50): (0.5^50) * (0.5^50) = 8.881784197001252e-16 Finally, we multiply the two results: 100891344545564193334812497256 * 8.881784197001252e-16 ≈ 0.0795892 So the probability of getting exactly 50 heads in 100 flips of a fair coin is approximately 0.0796 or 7.96%.,The probability of getting exactly 50 heads in 100 coin flips is approximately 7.9%. This can be calculated using the binomial probability formula: P(X = 50) = C(100, 50) * (0.5)^50 * (0.5)^50 Where C(100, 50) is the number of combinations of choosing 50 heads from 100 flips and 0.5 represents the probability of getting heads in each flip (since it's a fair coin). P(X = 50) ≈ 0.079153 Please note that this is an approximation, and the actual probability may slightly differ due to rounding errors., To calculate the probability of getting exactly 50 heads in 100 flips, we need to use the binomial probability formula: P(X = k) = C(n, k) * p^k * (1-p)^(n-k) Where: - P(X = k) is the probability of getting k successes (in this case, heads) - C(n, k) is the number of combinations of n items taken k at a time - p is the probability of success on any given trial (0.5 for a fair coin) - n is the number of trials (flips) - k is the number of successes (heads) In our case, n = 100, k = 50, and p = 0.5. First, let's calculate C(100, 50): C(100, 50) = 100! / (50! * (100 - 50)!) = 100! / (50! * 50!) We can simplify this expression by noticing that: 100! = 100 * 99 * 98 * ... * 2 * 1 = (100 * 99 * 98 * ... * 51) * (50 * 49 * ... * 2 * 1) = (100 * 99 * 98 * ... * 51) * 50! So, C(100, 50) = 100! / (50! * 50!) = (100 * 99 * 98 * ... * 51) / (50!) Now, let's plug these values into the binomial probability formula: P(X = 50) = C(100, 50) * p^50 * (1-p)^(100-50) = ((100 * 99 * 98 * ... * 51) / (50!)) * (0.5)^50 * (0.5)^50 = (1.0061445179124183e+29) * (0.0000009094947017729282) After calculating the result, we get: P(X = 50) ≈ 0.09094947017729282 So, the probability of getting exactly 50 heads in 100 flips of a fair coin is approximately 9.09%.