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This quiz was administered in-person. Students were allowed a cheat sheet. Students had 20 minutes to work on the quiz.

This quiz covered Lectures 18-21 of the Winter 2025 offering of DSC 10.

Problem 1

You want to estimate the proportion of all flower fields in the world that use fertilizer. You will create a 68% confidence interval for this proportion, based on a sample of flower fields. You want your confidence interval to have a width of at most 0.01. Using the fact that the standard deviation of any dataset of 0s and 1s is no more than 0.5, calculate the minimum sample size required. Give your answer as an integer.

Problem 2

A researcher collects data on many flowers, some of which have been treated with fertilizer and some of which have not (untreated). The researcher wants to determine whether there is a relationship between fertilizer use and the height of the flowers.

Problem 2.1

Choose the correct formulation of the null and alternative hypotheses.

• Null: There is a difference in the average height of treated and untreated flowers. Alt: There is no difference in the average height of treated and untreated flowers.

• Null: There is no difference in the average height of treated and untreated flowers. Alt: There is a difference in the average height of treated and untreated flowers.
  

• Null: There is no difference in the average height of treated and untreated flowers. Alt: Treated flowers grow taller,on average, than untreated flowers.

• Null: Fertilizer use has no impact on flower height. Alt: Fertilizer use causes taller flowers.

Problem 2.2

True or False: We could use a permutation test to test these hypotheses.

• True

• False

Problem 3

The DataFrame flowers contains information on a sample of flower fields. Each row corresponds to a different flower field. The "Fertilizer" column contains Boolean values corresponding to whether each flower field uses fertilizer.

You wonder if it could be the case that flower fields are fertilized at random, with each flower field having an 80% chance of being fertilized, independently of all others. You decide to do a hypothesis test to determine whether this could or could not be the case, testing at the $p=0.05$ significance level. You will use as your test statistic the absolute difference between 0.8 and the proportion of fertilized flower fields.

Problem 3.1

Complete the implementation of the function one_stat, which calculates one simulated value of this test statistic, under the assumptions of the null hypothesis. Note that the optional argument p in np.random.choice specifies the probabilities with which each element is chosen (here, there is a 0.8 probability of selecting True).

def one_stat():
    sample_size = __(a)__
    random_choice = (np.random.choice([True, False], 
                     sample_size, p=[0.8, 0.2]))
    return __(b)__

Problem 3.2

Complete the implementation of the function one_stat_differently, which also calculates one simulated value of this test statistic under the null.

def one_stat_differently():
    multi = np.random.multinomial(__(c)__)
    return __(d)__

Problem 3.3

Fill in the blanks to calculate 10,000 simulated values of the test statistic and collect them in an array called many_stats. You can use the functions you’ve already written to help you.

        many_stats = __(e)__
        for i in np.arange(10000):
            many_stats = __(f)__

Problem 3.4

Suppose that the observed value of the test statistic is 0.04. What do we conclude?

• Reject the null hypothesis.

• Accept the null hypothesis.

• Fail to reject the null hypothesis.

• Not enough information.

Problem 3.5

True or False: We could construct a confidence interval to test these hypotheses.

• True

• False

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