Statistical Analysis of Data Sets and Probabilities

Substance Concentration Data Grouping

The content of a substance in a liquid is given with an accuracy of 5 milligrams per liter. As such, the data can take values from this sequence:

• 120
• 125
• 130
• 135
• 140
• 145

Grouping Amplitude and Frequency Table Construction

The problem states a data set ranging between 110 and 245. However, the provided calculation for the number of steps, (247.7 - 107.5) / 25 = 5.6, implies a wider range. This calculation indicates that 6 steps (intervals) of 25 units each are needed. Since 6 intervals of 25 units cover 150 units, the apparent limits can be started with a round number to accommodate the required range.

Probability and Independence of Events

Disease Treatment Effectiveness and Repeat Offenders

It has been tested for a disease treatment that was effective in 82% of cases. Of those affected, 26% were repeat offenders, and the treatment was effective in half of them.

Treatment Effectiveness and Repeat Offender Independence

Are the events E (treatment is effective) and R (the patient is a repeat offender) independent? Provide a rationale and interpret the results.

Given: P(E) = 0.82 and P(E|R) = 0.50. Since P(E) ≠ P(E|R), the events are not independent.

Blood Donor Probabilities

In a medical facility, 30% of blood donors are also platelet donors. There are twelve blood donors in the facility.

Probability of No Platelet Donors

What is the probability that none of the twelve blood donors donate platelets?

Probability: 0.0138

Probability of Meeting Platelet Donor Need

If three platelet donors are needed at that time, what is the probability that this need can be met with the twelve donors in the center?

Probability: 0.7472

Normal Distribution and Confidence Intervals

Girls' Weight Distribution Analysis

The weight of eight-year-old girls follows a normal distribution with a mean of 26.8 kg and a standard deviation of 6.84 kg.

Percentage of Girls Weighing Between 25 and 30 kg

What percentage of eight-year-old girls weigh between 25 and 30 kg?

Percentage: 0.2874 (or 28.74%)

Weight at the 62nd Percentile

What weight corresponds to a girl who is in the 62nd percentile?

Weight: 28.92 kg

Average Weight of Six Girls Below 25 kg

Interpret the probability that six girls of that age, on average, weigh less than 25 kg.

Probability: 0.2611

Patient Satisfaction Confidence Interval

An interval estimate is desired for the proportion of people attending a medical facility consultation who are satisfied with their treatment. A sample of 500 was taken, of which 58% were satisfied.

Calculating and Interpreting the 95% Confidence Interval

Find the confidence interval, using a 95% confidence level, and interpret it.

The 95% confidence interval is (0.5367, 0.6233). This means we are 95% confident that the true proportion of satisfied patients in the population lies between 53.67% and 62.33%.

Comparing 90% and 95% Confidence Intervals

Explain, without calculation, whether a 90% confidence interval would be wider or narrower than a 95% confidence interval.

A 90% confidence interval would be narrower than a 95% confidence interval. This is because a lower confidence level requires a smaller range to capture the true parameter, as it allows for a higher chance of the interval not containing the true value.