Statistics and Probability Answers

You took a quiz in Mathematics but your teacher did not tell you your exact answer was. Instead,he informed you that 20% of your classmates higher than you. What was your score?

If your IQ is 85, what percentage of people have a higher IQ than you? IQ mean is 100 and the standard deviation is 15.

Why standard deviation is the better measure of variability?

The probability that a salesman makes a sale on a visit to a prospective client is 0.2. What is
the probability that, in 2 visits, of:
i) making no sales
ii)making more than one sale

The new memory enhancer X recently gained a reputation in the market as the most effective memory enhancer ever produced by XY Pharma. The recorded average memory level among who had taken the enhancer was 95 with a standard deviation of 10. As a neuroscientist, you want to test the claim and conducted a study and measure the memory level of 50 participants. The average memory level of the participants came out to be 80. Is there enough evidence to support the claim about the new memory enhancer?

If a person tests positive for Tuberculosis (TB) and starts treatment immediately, it takes on average 6 months (183 days, with a standard deviation of 14 days) of treatment until they are completely TB free. To receive the positive TB test individuals may go for the tuberculin skin test which takes on average 2.5 days to receive these test results with a standard deviation of 0.5 days, after which they are immediately given TB treatment. If we are interested in the total time until a person is TB free from the time of the TB test, what is the mean of the distribution that describes this time from TB test until a TB-free status?

IN R studio

# 3. Rolling two dice.Experiment: Given two die which are biased to land on 3 with probability .5 and equally likely to land on all other numbers with probability .1.We roll these die.

Define a random variable "sum_dice" that calculates the sum of the numbers that show up on the two dice. ```{r}```

Run a simulation of 100000 rolls. Use this simulation to compute the empirical probability distribution of this random variable. Compare this with the theoretical probability distribution. ```{r}```

Compute the expected value and the variation of this random variable. ```{r}```

IN R studio

# 2. More Coin TossesExperiment: A coin toss has outcomes {H, T}, with P(H) = .6. We do independent tosses of the coin until we get a head.
Recall that we computed the sample space for this experiment in class, it has infinite number of outcomes.
Define a random variable "tosses_till_heads" that counts the number of tosses untilwe get a heads. ```{r}```
Use the replicate function, to run 100000 simulations of this random variable. ```{r}```

Use these simulations to estimate the probability of getting a head after 15 tosses. Compare this with the theoretical value computed in the lectures.```{r}```

Compute the probability of getting a head after 50 tosses. What do you notice?```{r}```

IN R STUDIO
1. Coin TossesExperiment: A coin toss has outcomes {H, T}, with P(H) = .6.
We do five independent tosses of this coin.
Recall that we have computed the sample space for this experiment in class.
a) Define a random variable "num_heads" that takes input a sample vector and outputs the number of heads. ```{r}```
Use the replicate function, to run 100000 simulations of this random variable. Use this information to compute the probability distribution of "num_heads". Compare this with the theoretical distribution computed in class.```{r}```

the probability it will rain on a given day in july is 15% .what is the probability that within one week it will rain no more than 3 days

if a person tests positive for Tuberculosis and starts treatment immediately it takes on avarage 6 months ( 183 days with a standard deviation of 14 days) of treatment until they are completely TB free .what is the probability that a person will be healthy and TB free in less than 160 days