SOLVE ALL QUESTION IN STEP BY STEP

1A company claims that 90% of its products meet quality standards. In a sample of 200 products, only 160 meet the standards. Test the claim at a 5% significance level.

2A study measures the heights of male and female students in a school. The heights of male students follow a normal distribution with a mean of 175 cm and a standard deviation of 6 cm. The heights of female students follow a normal distribution with a mean of 165 cm and a standard deviation of 5 cm. What is the probability that a randomly chosen male student is taller than a randomly chosen female student?

3A box contains 5 red balls and 3 blue balls. Two balls are randomly drawn without replacement. What is the probability of drawing two red balls?

4A factory produces light bulbs, and the number of defective bulbs produced each day follows a Poisson distribution with a mean of 2 defects per day. What is the probability that exactly 4 defective bulbs will be produced tomorrow?

5In a clinical trial, 30% of patients experienced side effects from a new drug. If 10 patients are randomly selected, what is the probability that exactly 3 of them experienced side effects?

6A car rental company knows that the time (in hours) a customer will rent a car follows a uniform distribution between 2 and 8 hours. What is the probability that a randomly chosen customer will rent a car for more than 4 hours?

7A teacher wants to know if there is a significant difference in test scores between two groups of students. Group A has 50 students with a mean score of 85 and a standard deviation of 10. Group B has 45 students with a mean score of 78 and a standard deviation of 8. Perform a hypothesis test at a 1% significance level.

8A factory produces widgets, and the weight of each widget follows a normal distribution with a mean of 500 grams and a standard deviation of 20 grams. What is the probability that the average weight of 25 randomly chosen widgets is less than 490 grams?

9A survey is conducted to estimate the proportion of people in a city who support a new tax proposal. The sample size is 500, and the proportion in favor of the tax proposal is 0.55. Calculate the 95% confidence interval for the population proportion.

10Two dice are rolled. What is the probability that the sum of the two numbers rolled is 7?

11The waiting time (in minutes) for a bus to arrive at a stop follows an exponential distribution with a rate parameter of 0.2. What is the probability that a passenger will wait more than 10 minutes for the bus?

12A factory produces computer chips, and the number of defective chips produced each day follows a geometric distribution with a success probability of 0.1. What is the probability that the first defective chip occurs on the third production?

13A university offers two statistics courses: Course A and Course B. The success rates of students in the courses are 70% and 80%, respectively. A student randomly selects one of the courses. Given that the student passed the course, what is the probability that the student took Course B?

14In a large population, the heights of individuals follow a normal distribution with a mean of 170 cm and a standard deviation of 8 cm. What is the minimum height of the top 5% tallest individuals in the population?

15A coin is flipped 100 times, and it lands heads up 60 times. Test the hypothesis that the coin is fair at a 1% significance level.

16A company sells smartphones, and the lifetime of each phone follows a Weibull distribution with shape parameter 2 and scale parameter 800 days. What is the probability that a randomly chosen phone will last between 700 and 900 days?

17The lifetimes of light bulbs are normally distributed with a mean of 1000 hours and a standard deviation of 100 hours. What is the probability that the range between the lifetimes of the longest 5% and shortest 5% of bulbs is less than 200 hours?

18A study measures the reaction times (in milliseconds) of two groups of participants. Group A has a mean reaction time of 250 ms with a standard deviation of 30 ms, while Group B has a mean reaction time of 270 ms with a standard deviation of 25 ms. Perform a hypothesis test to determine if there is a significant difference in reaction times between the two groups at a 5% significance level.

19A random sample of 50 students is selected from a school, and their average test score is 75 with a standard deviation of 10. Construct a 99% confidence interval for the population mean test score.

20A casino offers a card game in which the player draws cards one by one from a standard deck of 52 cards. The game ends when the first ace is drawn. What is the probability that the game ends on the third card drawn?