Exercises: Normal approximation to the binomial distribution

Given the standard normal distribution $Z \sim N(0, 1)$, use the probability table to calculate the cumulative area to the left of the following values. Graphically represent the situation in part a.

1. a) $P(Z \le 1,32)$
2. b) $P(Z < 2,05)$
3. c) $P(Z \le 0,48)$

In a $Z \sim N(0, 1)$ distribution, we know that the total area under the curve is equal to $1$. Use the properties of symmetry and the complementary event to find the following probabilities:

1. a) $P(Z > 1,15)$
2. b) $P(Z \ge 0,64)$
3. c) $P(Z > 2,80)$

A study on the time 16-year-old teenagers spend using a well-known social network per day indicates that it follows a normal distribution with a mean of $\mu = 120$ minutes and a standard deviation of $\sigma = 20$ minutes.

If we choose a young person at random, calculate the probability that they use the social network for less than $150$ minutes. To do this, standardize the variable $X$ and use the $N(0, 1)$ table.

The scores of an Applied Mathematics for Social Sciences exam follow a $N(5,2; 1,5)$ distribution. If a score higher than $8$ is required to obtain a special mention, what is the probability of a student achieving it?

Hint: Remember that $P(X > k) = 1 - P(X \le k)$ after standardizing.

Find the value of the abscissa $k$ (the value of the variable on the horizontal axis) in a standard normal distribution $Z \sim N(0, 1)$ knowing that:

1. a) $P(Z \le k) = 0,8413$
2. b) $P(Z \le k) = 0,9750$
3. c) $P(Z \le k) = 0,5000$

The weight of apples from a harvest follows a normal distribution with a mean of $150$ ounces and a standard deviation of $15$ ounces. If apples weighing between $135$ and $165$ ounces are considered "type A apples", calculate the probability that a randomly chosen apple is type A.

A basketball free-throw shooter has a probability of success of $p = 0,8$. If they take $n = 100$ shots, we want to know if we can approximate the binomial variable $X$ (number of successful shots) by a normal distribution.

1. a) Check if the conditions $n \cdot p \ge 5$ and $n \cdot q \ge 5$ are met.
2. b) If so, state what the parameters $\mu$ and $\sigma$ of the new normal distribution $X'$ would be.

A fair coin ($p = 0,5$) is flipped $400$ times. Calculate the probability of getting fewer than $190$ heads using the normal approximation to the binomial (without applying the continuity correction yet).

Step 1: Calculate the mean $\mu$ and the standard deviation $\sigma$.

Step 2: Standardize the value $X = 190$ and look it up in the table.

The time that high school juniors dedicate daily to doing homework follows a normal distribution with a mean of $\mu = 115$ minutes and a standard deviation of $\sigma = 25$ minutes. Calculate the probability that a randomly chosen student:

1. a) Dedicates between 90 and 140 minutes a day.
2. b) Dedicates more than 160 minutes.

A gaming peripherals brand has determined that the response time of its flagship mouse follows a $N(18, 4)$ milliseconds distribution. The marketing department wants to promote that their product is so fast that only $15\%$ of competitors' mice (which follow the same distribution) exceed a critical response time $k$. Find the value of this time $k$.

It is known that the weight of coffee packets from a certain factory follows a normal distribution. After a quality control check, it is observed that $5\%$ of the packets weigh less than $242$ ounces and $10\%$ weigh more than $255$ ounces. Find the mean $\mu$ and the standard deviation $\sigma$ of this population.

Two friends, Marcus and Sarah, study at different schools. At Marcus's school, the mean score on the Economics exam was $6,2$ with a standard deviation of $1,1$. At Sarah's, the mean was $6,8$ with a standard deviation of $1,5$. If Marcus obtained a $7,5$ and Sarah a $8,2$, and assuming that both populations follow a normal distribution, determine which of the two obtained a better relative score compared to their group.

A study on recycling habits states that $35\%$ of households in a city separate plastic correctly. If a random sample of $120$ households is selected:

1. a) Justify why this experiment can be approximated by a normal distribution.
2. b) Calculate the probability that, in this sample, between $40$ and $50$ households (inclusive) separate plastic.

A multiple-choice exam consists of $80$ questions, each with $4$ answer options of which only one is correct. If a student answers all the questions completely at random:

1. a) Determine the parameters of the normal distribution that approximates this variable.
2. b) Calculate the probability that they get at least $25$ correct answers.

The battery life of a smartphone model follows a $N(42, 6)$ hours distribution. The company wants to offer a replacement warranty if the battery lasts less than a certain time. What maximum duration should the company guarantee if it is only willing to replace $2,5\%$ of the batteries sold?

On a streaming platform, $15\%$ of users cancel their subscription before the end of the trial month. If $400$ new users have signed up this month, calculate, using the normal approximation with continuity correction, the probability that exactly $65$ users unsubscribe.

It is known that the charging time of the batteries of a new model of electric car follows a normal distribution $N(\mu, \sigma)$. After carrying out a market study, it is observed that 15% of the vehicles take less than 45 minutes to charge completely, while only 20% of the vehicles require more than 75 minutes.

Calculate the value of the mean $\mu$ and the standard deviation $\sigma$ of this distribution. Then, determine the probability that a randomly chosen vehicle takes between 50 and 60 minutes to charge.

In a prestigious esports tournament, the score obtained by the participants follows a normal distribution with a mean of 2500 points and a standard deviation of 400 points. The organization decides to award the "Elite Mention" only to the 2.5% of players with the highest scores.

1. a) From what score is this mention received?
2. b) If a group of friends has scores that lie in the central interval comprising 95% of the players in the tournament, between which two values do their scores range?

A courier company uses drones for its deliveries in urban centers. It has been proven that the flight time of these drones until the battery runs out follows a normal distribution with a mean of 42 minutes. If 97.72% of the drones fly for more than 30 minutes, what is the probability that a randomly chosen drone runs out of battery between 35 and 45 minutes of flight?

A study on reading habits in teenagers reveals that 70% of young people aged between 15 and 16 prefer digital format over paper. If we select a random sample of 400 teenagers:

1. a) Justify whether it is possible to approximate the distribution of the variable "number of teenagers who prefer digital format" using a normal distribution.
2. b) Calculate, using the previous approximation and applying the continuity correction, the probability that between 270 and 290 teenagers (inclusive) prefer the digital format.

A biased coin is flipped 500 times, where the probability of getting heads is 0.4. Calculate the probability of getting:

1. a) Exactly 200 heads.
2. b) More than 215 heads.
3. c) Fewer than 185 heads.

Note: Use the normal approximation with Yates's continuity correction.

5% of the parts manufactured by a high-precision machine are defective. If they are packed in boxes of 200 units, calculate the probability that in a box:

1. a) There are exactly 10 defective parts.
2. b) There are more than 15 defective parts.
3. c) The number of defective parts is less than or equal to 5.

Argue whether the results would be significantly different if the continuity correction were not applied.

In a multiple-choice exam with 100 questions, each with 4 answer options (only one correct), a student answers all the questions at random.

1. a) What is the probability that they get more than 30 questions right?
2. b) If at least 40 correct answers are needed to pass the exam, what is the probability that the student passes purely by chance?
3. c) Determine the interval $(np - 2\sigma, np + 2\sigma)$ and interpret its meaning in this context.

A gym chain knows that 20% of its registered members never visit the facilities. If there are 600 registered members in a specific gym:

1. a) Calculate the probability that the number of members who never visit is between 110 and 135 inclusive.
2. b) If the gym decides to launch a motivation campaign for these members if the number of "inactive" members exceeds 140, what is the probability that they will have to launch this campaign?
3. c) What is the maximum number of inactive members expected to be found with a probability of 90%?